Exponent Rules
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Exponential Equations (Like Bases)
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(Exponential Equations (Substitution
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[{"Name":"Exponent Rules","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Intro","Duration":"4m 24s","ChapterTopicVideoID":8006,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8006.jpeg","UploadDate":"2020-09-30T14:40:47.2430000","DurationForVideoObject":"PT4M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.155","Text":"In this clip, I\u0027m going to be talking about rules of exponents."},{"Start":"00:04.155 ","End":"00:06.360","Text":"Before I get into this properly,"},{"Start":"00:06.360 ","End":"00:09.660","Text":"I\u0027d like to start with a brief review or introduction."},{"Start":"00:09.660 ","End":"00:11.460","Text":"What is an exponent?"},{"Start":"00:11.460 ","End":"00:15.690","Text":"I\u0027ll start with an example 2^5,"},{"Start":"00:15.690 ","End":"00:17.100","Text":"that\u0027s how we pronounce it."},{"Start":"00:17.100 ","End":"00:20.130","Text":"We write the 5 here to the top right,"},{"Start":"00:20.130 ","End":"00:22.050","Text":"and it\u0027s a superscript,"},{"Start":"00:22.050 ","End":"00:29.160","Text":"2^5, and the way we define it is we multiply 2 with itself."},{"Start":"00:29.160 ","End":"00:34.665","Text":"They are altogether 5 times the 2 or 5 factors,"},{"Start":"00:34.665 ","End":"00:40.355","Text":"so I indicate this sometimes with curly brackets and I say there\u0027s 5 times."},{"Start":"00:40.355 ","End":"00:42.480","Text":"Of course, there\u0027s a numerical answer,"},{"Start":"00:42.480 ","End":"00:44.825","Text":"2 times 2 is 32."},{"Start":"00:44.825 ","End":"00:49.715","Text":"This part is called the base and this is the exponent."},{"Start":"00:49.715 ","End":"00:51.140","Text":"I\u0027ll just say that in general,"},{"Start":"00:51.140 ","End":"00:54.694","Text":"when we have some number to the power of some other number,"},{"Start":"00:54.694 ","End":"00:57.439","Text":"this part here is called the base,"},{"Start":"00:57.439 ","End":"01:01.385","Text":"and this part is called the exponent or the power."},{"Start":"01:01.385 ","End":"01:02.690","Text":"When we pronounce it,"},{"Start":"01:02.690 ","End":"01:06.170","Text":"we say a^n,"},{"Start":"01:06.170 ","End":"01:07.985","Text":"2^5, but 5 is the exponent."},{"Start":"01:07.985 ","End":"01:11.320","Text":"Another example, 3^2,"},{"Start":"01:11.320 ","End":"01:13.665","Text":"also called 3^2,"},{"Start":"01:13.665 ","End":"01:15.330","Text":"is 3 times 3."},{"Start":"01:15.330 ","End":"01:17.070","Text":"There\u0027s only 2 of them,"},{"Start":"01:17.070 ","End":"01:19.035","Text":"and this is equal to 9."},{"Start":"01:19.035 ","End":"01:22.725","Text":"In fact, we can even take just 1,"},{"Start":"01:22.725 ","End":"01:28.630","Text":"like 4^1, means we just take 1 factor 4 and it\u0027s just 4."},{"Start":"01:28.630 ","End":"01:31.715","Text":"That\u0027s the review of what an exponent is."},{"Start":"01:31.715 ","End":"01:37.370","Text":"I\u0027d also like to remind you of something called order of operations."},{"Start":"01:37.370 ","End":"01:41.030","Text":"Taking a power or the exponent of something gets an operation,"},{"Start":"01:41.030 ","End":"01:45.605","Text":"and it is done before multiplication and division,"},{"Start":"01:45.605 ","End":"01:47.645","Text":"before addition and subtraction."},{"Start":"01:47.645 ","End":"01:49.460","Text":"Just give you an example."},{"Start":"01:49.460 ","End":"01:54.735","Text":"Suppose I write 3 times 2^4,"},{"Start":"01:54.735 ","End":"01:59.060","Text":"even though multiplication comes first from left to right,"},{"Start":"01:59.060 ","End":"02:01.145","Text":"we do the exponent first,"},{"Start":"02:01.145 ","End":"02:05.550","Text":"and we say it\u0027s 3 times 2^4 which is"},{"Start":"02:05.550 ","End":"02:10.710","Text":"2 times 2 times 2 times 2 is 16, and that equals 48."},{"Start":"02:10.710 ","End":"02:13.310","Text":"I\u0027ll just say it, but I won\u0027t write it,"},{"Start":"02:13.310 ","End":"02:16.700","Text":"what not to do is to say 3 times 2 is 6,"},{"Start":"02:16.700 ","End":"02:18.650","Text":"and then take 6^4."},{"Start":"02:18.650 ","End":"02:21.350","Text":"I don\u0027t even know what that is,"},{"Start":"02:21.350 ","End":"02:25.805","Text":"maybe 1,296, but it\u0027s not the answer."},{"Start":"02:25.805 ","End":"02:31.860","Text":"Another example is suppose I take -3^4,"},{"Start":"02:32.270 ","End":"02:35.570","Text":"there\u0027s a power and there\u0027s a minus,"},{"Start":"02:35.570 ","End":"02:36.650","Text":"which is a subtraction."},{"Start":"02:36.650 ","End":"02:39.770","Text":"The power is done before the subtraction."},{"Start":"02:39.770 ","End":"02:43.185","Text":"This is 3^4 is 81,"},{"Start":"02:43.185 ","End":"02:44.760","Text":"3 times 3 times 3 times 3,"},{"Start":"02:44.760 ","End":"02:46.955","Text":"and the answer is -81."},{"Start":"02:46.955 ","End":"02:52.220","Text":"It is not correct to say -3 times -3 times -3 times -3."},{"Start":"02:52.220 ","End":"02:54.535","Text":"That would give you +81."},{"Start":"02:54.535 ","End":"02:56.965","Text":"This is a brief review,"},{"Start":"02:56.965 ","End":"03:00.835","Text":"I\u0027d just like to show you what rules of exponents look like."},{"Start":"03:00.835 ","End":"03:04.200","Text":"I found a picture of all the rules,"},{"Start":"03:04.200 ","End":"03:06.844","Text":"not necessarily going to do them in this order."},{"Start":"03:06.844 ","End":"03:14.360","Text":"Just want you to notice something that I could put a separating line here."},{"Start":"03:14.360 ","End":"03:18.740","Text":"Notice that these rules have roots in them."},{"Start":"03:18.740 ","End":"03:21.620","Text":"Exponents and roots belonged together,"},{"Start":"03:21.620 ","End":"03:27.615","Text":"so really this should have been called rules of exponents and roots."},{"Start":"03:27.615 ","End":"03:30.735","Text":"Roots are also called radicals."},{"Start":"03:30.735 ","End":"03:34.860","Text":"There\u0027s actually 1 more that\u0027s not in this diagram,"},{"Start":"03:34.860 ","End":"03:38.990","Text":"and that rule is a^n is always bigger than 0,"},{"Start":"03:38.990 ","End":"03:41.450","Text":"but we\u0027ll get to that later on."},{"Start":"03:41.450 ","End":"03:45.080","Text":"I just wanted to get an impression of what the rules look like."},{"Start":"03:45.080 ","End":"03:47.520","Text":"I almost forgot something very important."},{"Start":"03:47.520 ","End":"03:49.725","Text":"When we use rules of exponents,"},{"Start":"03:49.725 ","End":"03:53.975","Text":"then we\u0027re going to assume that a is bigger than 0,"},{"Start":"03:53.975 ","End":"03:57.370","Text":"both for exponents and for roots."},{"Start":"03:57.370 ","End":"03:59.510","Text":"Finally, I\u0027m often asked,"},{"Start":"03:59.510 ","End":"04:01.190","Text":"what are these rules good for?"},{"Start":"04:01.190 ","End":"04:03.215","Text":"I\u0027ll just say a few words."},{"Start":"04:03.215 ","End":"04:06.260","Text":"This will be useful in the future for all kinds of things in"},{"Start":"04:06.260 ","End":"04:09.140","Text":"mathematics if you learn calculus and even if not,"},{"Start":"04:09.140 ","End":"04:12.530","Text":"this is going to be indispensable for the section on"},{"Start":"04:12.530 ","End":"04:16.470","Text":"exponential equations and in general,"},{"Start":"04:16.470 ","End":"04:21.275","Text":"to simplifying and manipulating algebraic expressions with exponents."},{"Start":"04:21.275 ","End":"04:25.380","Text":"Let\u0027s get started on the rules themselves."}],"ID":8099},{"Watched":false,"Name":"Rules 1-2","Duration":"5m 4s","ChapterTopicVideoID":8007,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8007.jpeg","UploadDate":"2020-09-30T14:42:34.3700000","DurationForVideoObject":"PT5M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"Now let\u0027s get started with the rules."},{"Start":"00:02.550 ","End":"00:04.005","Text":"Rule number 1,"},{"Start":"00:04.005 ","End":"00:08.430","Text":"a^m times"},{"Start":"00:08.430 ","End":"00:16.440","Text":"a^n is equal to a^(m+n)."},{"Start":"00:16.440 ","End":"00:18.930","Text":"This rule is sometimes called the product rule."},{"Start":"00:18.930 ","End":"00:21.330","Text":"And I\u0027ll give an example."},{"Start":"00:21.330 ","End":"00:31.250","Text":"For the example, let\u0027s take 5^2 or 5 squared times 5^4 or 5 to the fourth."},{"Start":"00:31.250 ","End":"00:36.275","Text":"Notice the important thing is that it\u0027s the same 5, the same a."},{"Start":"00:36.275 ","End":"00:38.000","Text":"Now according to this rule,"},{"Start":"00:38.000 ","End":"00:40.820","Text":"if m is 2 and n is 4,"},{"Start":"00:40.820 ","End":"00:46.380","Text":"I should be able to say that this is equal to 5^(2+4)."},{"Start":"00:46.380 ","End":"00:51.310","Text":"Of course, 2 plus 4 is 6 so this should be 5^6."},{"Start":"00:51.310 ","End":"00:55.745","Text":"It looks strange because here we\u0027re doing a multiplication,"},{"Start":"00:55.745 ","End":"00:58.174","Text":"and yet here we have an addition."},{"Start":"00:58.174 ","End":"01:00.800","Text":"I\u0027ll show you why this is so,"},{"Start":"01:00.800 ","End":"01:04.760","Text":"5^2 is 5 times 5."},{"Start":"01:04.760 ","End":"01:10.894","Text":"I\u0027m putting it in brackets just for emphasis that there\u0027s 2 factors of 5"},{"Start":"01:10.894 ","End":"01:17.760","Text":"and 5^4 is 5 times 5 times 5 times 5."},{"Start":"01:17.760 ","End":"01:21.195","Text":"That would be 4 factors of 4."},{"Start":"01:21.195 ","End":"01:25.995","Text":"Now if I multiply 5 times 5 times 5 times 5 times 5,"},{"Start":"01:25.995 ","End":"01:32.305","Text":"I should get just 5 times 5 times 5 times 5 times 5 times 5."},{"Start":"01:32.305 ","End":"01:35.255","Text":"And if I count how many I have here,"},{"Start":"01:35.255 ","End":"01:37.115","Text":"I now have 6."},{"Start":"01:37.115 ","End":"01:42.200","Text":"You can see why this addition rule applies because although we\u0027re multiplying,"},{"Start":"01:42.200 ","End":"01:45.335","Text":"we\u0027re just adding to the number of factors,"},{"Start":"01:45.335 ","End":"01:46.820","Text":"number of times 5 appears,"},{"Start":"01:46.820 ","End":"01:49.100","Text":"it appears twice and it appears 4 times."},{"Start":"01:49.100 ","End":"01:51.545","Text":"We multiply it appears 6 times."},{"Start":"01:51.545 ","End":"01:57.755","Text":"So that is this rule and let\u0027s get onto the second rule."},{"Start":"01:57.755 ","End":"02:02.930","Text":"Otherwise these numbers are just for this course is not a standard rule number."},{"Start":"02:02.930 ","End":"02:06.965","Text":"The second one is called the quotient rule sometimes,"},{"Start":"02:06.965 ","End":"02:08.180","Text":"and it\u0027s similar to this,"},{"Start":"02:08.180 ","End":"02:09.830","Text":"but instead of multiplication,"},{"Start":"02:09.830 ","End":"02:11.135","Text":"we have a division."},{"Start":"02:11.135 ","End":"02:14.675","Text":"So a^m divided by a^n,"},{"Start":"02:14.675 ","End":"02:18.230","Text":"and you might be able to guess if multiplication becomes addition,"},{"Start":"02:18.230 ","End":"02:21.185","Text":"then division becomes subtraction."},{"Start":"02:21.185 ","End":"02:26.250","Text":"So it\u0027s a^(m minus n) and here\u0027s an example,"},{"Start":"02:26.250 ","End":"02:28.695","Text":"I\u0027ll take for my example,"},{"Start":"02:28.695 ","End":"02:35.570","Text":"4^5 divided by 4^3."},{"Start":"02:35.570 ","End":"02:37.835","Text":"According to the rule,"},{"Start":"02:37.835 ","End":"02:42.047","Text":"I would say that this is equal to 4^(5 minus 3)."},{"Start":"02:42.047 ","End":"02:47.940","Text":"5 minus 3 is 2, so it\u0027s 4^2."},{"Start":"02:47.940 ","End":"02:50.520","Text":"And let\u0027s explain why this is so."},{"Start":"02:50.520 ","End":"02:52.820","Text":"By the way, I didn\u0027t actually bother to compute"},{"Start":"02:52.820 ","End":"02:55.460","Text":"the numerical answers in these, that\u0027s less important."},{"Start":"02:55.460 ","End":"03:00.800","Text":"I mean, of course you could compute this 25 and this would be, I don\u0027t know,"},{"Start":"03:00.800 ","End":"03:05.060","Text":"625 and you could check this on the calculator and it would work,"},{"Start":"03:05.060 ","End":"03:07.310","Text":"but obviously it does work."},{"Start":"03:07.310 ","End":"03:10.580","Text":"Here also I\u0027m not going to actually compute 4^5."},{"Start":"03:10.580 ","End":"03:12.395","Text":"I don\u0027t need to do that."},{"Start":"03:12.395 ","End":"03:14.540","Text":"But I would like to explain why this rule"},{"Start":"03:14.540 ","End":"03:17.525","Text":"applies and we\u0027ll explain it through this example."},{"Start":"03:17.525 ","End":"03:20.825","Text":"It\u0027s because if I take 4^5,"},{"Start":"03:20.825 ","End":"03:26.400","Text":"I can write that as 4 times 4 times 4"},{"Start":"03:26.400 ","End":"03:32.370","Text":"times 4 times 4."},{"Start":"03:32.370 ","End":"03:33.390","Text":"5 of them,1, 2, 3, 4, 5."},{"Start":"03:33.390 ","End":"03:36.720","Text":"Now I divide by 4^3,"},{"Start":"03:36.720 ","End":"03:39.330","Text":"so I count fours 1,"},{"Start":"03:39.330 ","End":"03:41.430","Text":"2, 3 factors."},{"Start":"03:41.430 ","End":"03:43.634","Text":"Now things cancel."},{"Start":"03:43.634 ","End":"03:47.145","Text":"On the denominator I have 4 times 4 times 4,"},{"Start":"03:47.145 ","End":"03:49.380","Text":"on the numerator if I cut it off here,"},{"Start":"03:49.380 ","End":"03:51.885","Text":"it will be 4 times 4 times 4."},{"Start":"03:51.885 ","End":"03:55.205","Text":"I\u0027ll just put a 1 here when we don\u0027t have anything left, we write a 1."},{"Start":"03:55.205 ","End":"04:00.740","Text":"And so this is equal to 4 times 4 and this 4 times 4 is 4^2."},{"Start":"04:00.740 ","End":"04:02.945","Text":"So you can see when we have a division,"},{"Start":"04:02.945 ","End":"04:06.260","Text":"it\u0027s like I have 5 factors and 3 of them canceled."},{"Start":"04:06.260 ","End":"04:09.920","Text":"So what I was left with was 5 minus 3 factors,"},{"Start":"04:09.920 ","End":"04:11.465","Text":"which is 2 factors."},{"Start":"04:11.465 ","End":"04:14.150","Text":"We\u0027ll assume for the moment that this exponent is"},{"Start":"04:14.150 ","End":"04:17.105","Text":"bigger than this exponents so it will make sense."},{"Start":"04:17.105 ","End":"04:19.460","Text":"Before we move on to rule 3,"},{"Start":"04:19.460 ","End":"04:23.615","Text":"I\u0027d like to say something general that applies to all of the rules."},{"Start":"04:23.615 ","End":"04:28.070","Text":"We\u0027re going to make an assumption that the bases in this case we have a base a,"},{"Start":"04:28.070 ","End":"04:29.945","Text":"but in some of the rules we have a and b."},{"Start":"04:29.945 ","End":"04:33.575","Text":"We\u0027re going to make an assumption that when we have an exponent,"},{"Start":"04:33.575 ","End":"04:36.995","Text":"in other words, we have a base to the power of,"},{"Start":"04:36.995 ","End":"04:40.115","Text":"let\u0027s call it the exponent, or power,"},{"Start":"04:40.115 ","End":"04:44.030","Text":"will make the assumption that the base is positive,"},{"Start":"04:44.030 ","End":"04:45.950","Text":"base bigger than 0."},{"Start":"04:45.950 ","End":"04:50.000","Text":"I\u0027ll explain at the end after the rules why we assume this."},{"Start":"04:50.000 ","End":"04:52.130","Text":"In all that follows here,"},{"Start":"04:52.130 ","End":"04:58.005","Text":"I would take a as positive and in all the rules we assume that the base is positive."},{"Start":"04:58.005 ","End":"04:59.780","Text":"As I said, if those who care,"},{"Start":"04:59.780 ","End":"05:01.835","Text":"I\u0027ll give a reason at the end."},{"Start":"05:01.835 ","End":"05:05.190","Text":"Onto the next rule."}],"ID":8100},{"Watched":false,"Name":"Rules 3-5","Duration":"6m 44s","ChapterTopicVideoID":8008,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8008.jpeg","UploadDate":"2020-09-30T14:43:22.5500000","DurationForVideoObject":"PT6M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.835","Text":"The next rule, and it happens to be number 3,"},{"Start":"00:02.835 ","End":"00:04.500","Text":"is the power rule,"},{"Start":"00:04.500 ","End":"00:07.260","Text":"sometimes called, and it goes as follows."},{"Start":"00:07.260 ","End":"00:12.446","Text":"A^m, and then to the power of"},{"Start":"00:12.446 ","End":"00:18.480","Text":"n is equal to a^m times n. I\u0027ll add a dot here."},{"Start":"00:18.480 ","End":"00:20.670","Text":"You can see this is multiplication,"},{"Start":"00:20.670 ","End":"00:22.695","Text":"and I\u0027ll explain with an example."},{"Start":"00:22.695 ","End":"00:29.955","Text":"I\u0027ll take (4^2)^3."},{"Start":"00:29.955 ","End":"00:31.630","Text":"According to the rule,"},{"Start":"00:31.630 ","End":"00:38.035","Text":"it\u0027s says that this is equal to 4^2 times 3,"},{"Start":"00:38.035 ","End":"00:41.155","Text":"not 2^3, but 2 times 3,"},{"Start":"00:41.155 ","End":"00:43.985","Text":"which of course is 4^6."},{"Start":"00:43.985 ","End":"00:47.975","Text":"Now, like I said that numerically it doesn\u0027t really matter."},{"Start":"00:47.975 ","End":"00:51.245","Text":"But if you do want to check it 4^2 is 16,"},{"Start":"00:51.245 ","End":"00:56.165","Text":"this say 16^3 is the same as 4^6,"},{"Start":"00:56.165 ","End":"01:01.385","Text":"and if you check both this on the calculator, you get 4,096."},{"Start":"01:01.385 ","End":"01:04.805","Text":"But like I said, this is not what interests me."},{"Start":"01:04.805 ","End":"01:09.725","Text":"I\u0027d rather explain why this rule is so in general. Here\u0027s why."},{"Start":"01:09.725 ","End":"01:13.190","Text":"If I write 4^2 cubed, I can write 4^2,"},{"Start":"01:13.190 ","End":"01:17.775","Text":"4^2 is 4 times 4."},{"Start":"01:17.775 ","End":"01:19.470","Text":"If I cube something,"},{"Start":"01:19.470 ","End":"01:20.580","Text":"I cube this thing,"},{"Start":"01:20.580 ","End":"01:23.210","Text":"it means I take it 3 times and multiply."},{"Start":"01:23.210 ","End":"01:24.875","Text":"So I have 4 times 4,"},{"Start":"01:24.875 ","End":"01:26.675","Text":"I have another 4 times 4,"},{"Start":"01:26.675 ","End":"01:28.735","Text":"and I have yet another 4 times 4,"},{"Start":"01:28.735 ","End":"01:31.750","Text":"altogether I have 3 of them,"},{"Start":"01:31.750 ","End":"01:33.935","Text":"and I multiply them all together."},{"Start":"01:33.935 ","End":"01:36.155","Text":"Now, how many 4s do I get?"},{"Start":"01:36.155 ","End":"01:37.921","Text":"Well, I can just count and say,"},{"Start":"01:37.921 ","End":"01:39.639","Text":"yes, 1, 2,"},{"Start":"01:39.639 ","End":"01:41.068","Text":"3, 4,"},{"Start":"01:41.068 ","End":"01:43.220","Text":"5, 6 by counting them."},{"Start":"01:43.220 ","End":"01:44.765","Text":"But how did I get to 6?"},{"Start":"01:44.765 ","End":"01:49.070","Text":"Is because each one of these contains 2 each group,"},{"Start":"01:49.070 ","End":"01:50.765","Text":"and I have 3 groups of 2,"},{"Start":"01:50.765 ","End":"01:55.220","Text":"so that this 6 really is 2 times 3,"},{"Start":"01:55.220 ","End":"01:58.190","Text":"which is what we wrote here."},{"Start":"01:58.190 ","End":"02:01.025","Text":"I want to continue a bit further with this rule,"},{"Start":"02:01.025 ","End":"02:03.320","Text":"and note in this case for example,"},{"Start":"02:03.320 ","End":"02:08.170","Text":"that if someone had said to me what is (4^3)^2?"},{"Start":"02:08.170 ","End":"02:10.490","Text":"In other words, the exponents in a different order,"},{"Start":"02:10.490 ","End":"02:17.105","Text":"what I would\u0027ve got is 4^3 times 2 which is also 4^6."},{"Start":"02:17.105 ","End":"02:22.225","Text":"It\u0027s not a coincidence because 2 times 3 is the same as 3 times 2."},{"Start":"02:22.225 ","End":"02:31.885","Text":"What this comes down to is that I can write an extra equals in here, and write (a^n)^m."},{"Start":"02:31.885 ","End":"02:33.530","Text":"This is a bonus path,"},{"Start":"02:33.530 ","End":"02:36.095","Text":"and this is actually doesn\u0027t appear in most of the books,"},{"Start":"02:36.095 ","End":"02:38.150","Text":"but I find it to be very useful."},{"Start":"02:38.150 ","End":"02:41.480","Text":"It will come in useful that if you have a power of a power,"},{"Start":"02:41.480 ","End":"02:43.080","Text":"you can change the order,"},{"Start":"02:43.080 ","End":"02:44.360","Text":"and the rule itself says,"},{"Start":"02:44.360 ","End":"02:45.980","Text":"if you have a power of a power,"},{"Start":"02:45.980 ","End":"02:48.445","Text":"you just multiply the powers."},{"Start":"02:48.445 ","End":"02:50.400","Text":"That\u0027s rule number 3."},{"Start":"02:50.400 ","End":"02:51.765","Text":"Let\u0027s get onto the next one."},{"Start":"02:51.765 ","End":"02:55.140","Text":"In fact, I want to bring 2 rules at once."},{"Start":"02:55.140 ","End":"02:57.770","Text":"Together they\u0027re called the expanded power rule."},{"Start":"02:57.770 ","End":"02:59.180","Text":"The name is not important."},{"Start":"02:59.180 ","End":"03:03.980","Text":"One of them says that a times"},{"Start":"03:03.980 ","End":"03:10.715","Text":"b^n equals a^n times b^n."},{"Start":"03:10.715 ","End":"03:13.970","Text":"Note that up to now we\u0027ve always had the same base a,"},{"Start":"03:13.970 ","End":"03:17.005","Text":"this time we have 2 different bases a and b."},{"Start":"03:17.005 ","End":"03:18.830","Text":"This one relates to the product,"},{"Start":"03:18.830 ","End":"03:20.000","Text":"and I\u0027ll just write the other one,"},{"Start":"03:20.000 ","End":"03:21.320","Text":"and I\u0027ll explain them both."},{"Start":"03:21.320 ","End":"03:23.120","Text":"The other is very similar to this,"},{"Start":"03:23.120 ","End":"03:26.180","Text":"but everywhere you see a multiplication we put a division."},{"Start":"03:26.180 ","End":"03:33.545","Text":"That a over b^n is a^n over b^n."},{"Start":"03:33.545 ","End":"03:37.580","Text":"Now, let\u0027s give an example of each which will also explain the logic behind it."},{"Start":"03:37.580 ","End":"03:45.935","Text":"For this one, I\u0027ll take the example that 2 times 5^3 according to the rule,"},{"Start":"03:45.935 ","End":"03:53.225","Text":"would say that it\u0027s the same as 2^3 times 5^3."},{"Start":"03:53.225 ","End":"03:57.945","Text":"I\u0027d actually like to the numerical value in this case because it\u0027s easy,"},{"Start":"03:57.945 ","End":"04:05.640","Text":"2 times 5^3 is 10^3 which is 1,000,"},{"Start":"04:05.640 ","End":"04:07.860","Text":"and 2^3 is 8,"},{"Start":"04:07.860 ","End":"04:11.040","Text":"and 5^3 is 125."},{"Start":"04:11.040 ","End":"04:16.170","Text":"Indeed 1,000 is 8 times the 125."},{"Start":"04:16.170 ","End":"04:17.610","Text":"This is 1000,"},{"Start":"04:17.610 ","End":"04:20.430","Text":"and this time this is also 1,000."},{"Start":"04:20.430 ","End":"04:25.880","Text":"This checks, but as I said the numerical computations are less important."},{"Start":"04:25.880 ","End":"04:28.325","Text":"Let\u0027s explain the logic behind this."},{"Start":"04:28.325 ","End":"04:34.665","Text":"Well, 2 times 5 in brackets^3 means that I take 2 times 5,"},{"Start":"04:34.665 ","End":"04:36.765","Text":"and I\u0027ll take 3 such factors,"},{"Start":"04:36.765 ","End":"04:38.235","Text":"again 2 times 5,"},{"Start":"04:38.235 ","End":"04:40.455","Text":"and again 2 times 5."},{"Start":"04:40.455 ","End":"04:42.150","Text":"I\u0027ll put the little dots in between,"},{"Start":"04:42.150 ","End":"04:44.600","Text":"so really know it\u0027s multiplication."},{"Start":"04:44.600 ","End":"04:46.490","Text":"On the other hand, if I write it,"},{"Start":"04:46.490 ","End":"04:49.160","Text":"I can expand this out without the brackets,"},{"Start":"04:49.160 ","End":"04:54.200","Text":"and say it\u0027s 2 times 5 times 2 times 5 times 2 times 5."},{"Start":"04:54.200 ","End":"04:57.800","Text":"But since the order of multiplication makes no difference,"},{"Start":"04:57.800 ","End":"04:59.480","Text":"I can write all the 2s first,"},{"Start":"04:59.480 ","End":"05:01.600","Text":"2 times 2 times 2,"},{"Start":"05:01.600 ","End":"05:05.545","Text":"and then times 5 times 5 times 5."},{"Start":"05:05.545 ","End":"05:07.565","Text":"I think you can see where this is heading."},{"Start":"05:07.565 ","End":"05:10.580","Text":"Because this bit is 2^3."},{"Start":"05:10.580 ","End":"05:12.110","Text":"Let\u0027s get a bit more space here."},{"Start":"05:12.110 ","End":"05:14.690","Text":"This is 2^3 because there\u0027s 1,"},{"Start":"05:14.690 ","End":"05:16.220","Text":"2, 3 times the 2,"},{"Start":"05:16.220 ","End":"05:20.160","Text":"and this is 5^3 because again, 1, 2,"},{"Start":"05:20.160 ","End":"05:24.930","Text":"3 times, and this logic works for a and b in general."},{"Start":"05:24.930 ","End":"05:27.020","Text":"But this is a particular case,"},{"Start":"05:27.020 ","End":"05:30.395","Text":"but it proves the general case because it\u0027s all the same logic."},{"Start":"05:30.395 ","End":"05:33.215","Text":"Now, let\u0027s take an example of Rule Number 5."},{"Start":"05:33.215 ","End":"05:36.815","Text":"I think 2 and 5 are good numbers of work for me here also."},{"Start":"05:36.815 ","End":"05:39.730","Text":"In fact, I\u0027ll take the same n equals 3."},{"Start":"05:39.730 ","End":"05:45.555","Text":"Rule Number 5 would say that 2 divided by 5, or 2/5^3,"},{"Start":"05:45.555 ","End":"05:54.075","Text":"would equal 2^3 over 5^3 which means 2^3 over 5^3 when it\u0027s power of 3."},{"Start":"05:54.075 ","End":"05:56.670","Text":"You can compute it on the calculator,"},{"Start":"05:56.670 ","End":"05:59.195","Text":"this time I won\u0027t, I would just explain the logic."},{"Start":"05:59.195 ","End":"06:01.745","Text":"Remember, when we multiply fractions,"},{"Start":"06:01.745 ","End":"06:04.080","Text":"we multiply the numerator separately,"},{"Start":"06:04.080 ","End":"06:05.820","Text":"and the denominator separately."},{"Start":"06:05.820 ","End":"06:10.155","Text":"So 2/5^3 is 2/5 times 2/5 times 2/5."},{"Start":"06:10.155 ","End":"06:13.215","Text":"I\u0027ve counted 1, 2, 3 factors."},{"Start":"06:13.215 ","End":"06:17.435","Text":"Then because of the rules of multiplying fractions,"},{"Start":"06:17.435 ","End":"06:19.100","Text":"we multiply the numerators,"},{"Start":"06:19.100 ","End":"06:22.880","Text":"and say this is 2 times 2 times 2, and on denominator,"},{"Start":"06:22.880 ","End":"06:25.990","Text":"we have 5 times 5 times 5,"},{"Start":"06:25.990 ","End":"06:27.440","Text":"and this just gives us,"},{"Start":"06:27.440 ","End":"06:29.030","Text":"if we gather together the 2s,"},{"Start":"06:29.030 ","End":"06:30.860","Text":"this is 2^3,"},{"Start":"06:30.860 ","End":"06:33.730","Text":"and the 5 is 5^3."},{"Start":"06:33.730 ","End":"06:35.840","Text":"This shows why this works,"},{"Start":"06:35.840 ","End":"06:37.700","Text":"and it works in general."},{"Start":"06:37.700 ","End":"06:40.186","Text":"So that\u0027s, we\u0027ve got up to 5,"},{"Start":"06:40.186 ","End":"06:44.100","Text":"that\u0027s halfway, we got 5 more rules to go."}],"ID":8101},{"Watched":false,"Name":"Rules 6-8","Duration":"7m 12s","ChapterTopicVideoID":8009,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8009.jpeg","UploadDate":"2020-09-30T14:44:27.8700000","DurationForVideoObject":"PT7M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.305","Text":"Rule number 6 says the following, a^0=1."},{"Start":"00:07.305 ","End":"00:11.150","Text":"In actual fact, this is more of a definition than the rule."},{"Start":"00:11.150 ","End":"00:13.290","Text":"I\u0027ll write down the word definition."},{"Start":"00:13.290 ","End":"00:17.250","Text":"Here, we\u0027re defining what it means to take a number to the power of"},{"Start":"00:17.250 ","End":"00:22.785","Text":"0 because it doesn\u0027t make sense the usual way of multiplying a\u0027s together."},{"Start":"00:22.785 ","End":"00:26.240","Text":"It only makes sense for positive numbers, positive integers."},{"Start":"00:26.240 ","End":"00:29.975","Text":"We\u0027ll take this as a definition that any number to the power of 0 is 1."},{"Start":"00:29.975 ","End":"00:31.815","Text":"When I say any number,"},{"Start":"00:31.815 ","End":"00:35.760","Text":"I\u0027ll just remind you that the bases are always positive."},{"Start":"00:35.760 ","End":"00:37.710","Text":"Not that we\u0027re going to use that here,"},{"Start":"00:37.710 ","End":"00:40.895","Text":"but any positive number to the power of 0 is 1."},{"Start":"00:40.895 ","End":"00:43.895","Text":"Why did we decide to define it as 1?"},{"Start":"00:43.895 ","End":"00:47.165","Text":"It because we want all the rules to still work."},{"Start":"00:47.165 ","End":"00:51.170","Text":"For example, rule number 2, if you remember,"},{"Start":"00:51.170 ","End":"00:59.375","Text":"says that a^m minus n is a^m over a^n."},{"Start":"00:59.375 ","End":"01:01.700","Text":"If I take, for example,"},{"Start":"01:01.700 ","End":"01:03.185","Text":"a is 2,"},{"Start":"01:03.185 ","End":"01:06.125","Text":"I would get that 2 to the power of,"},{"Start":"01:06.125 ","End":"01:09.530","Text":"doesn\u0027t really matter, let\u0027s say 3 minus 3,"},{"Start":"01:09.530 ","End":"01:15.075","Text":"according to this rule, would be 2^3 over 2^3,"},{"Start":"01:15.075 ","End":"01:16.705","Text":"which is 8 over 8,"},{"Start":"01:16.705 ","End":"01:18.350","Text":"which doesn\u0027t really matter what it is,"},{"Start":"01:18.350 ","End":"01:21.815","Text":"it\u0027s going to equal 1 because same numerator and denominator."},{"Start":"01:21.815 ","End":"01:24.630","Text":"On the other hand, 3 minus 3 is 0."},{"Start":"01:24.630 ","End":"01:28.050","Text":"This would tell us that 2^0=1."},{"Start":"01:28.050 ","End":"01:32.860","Text":"Instead of 2, we could take any other number positive, that is."},{"Start":"01:32.860 ","End":"01:38.225","Text":"We now have exponents which can be 0 as well as positive integers."},{"Start":"01:38.225 ","End":"01:42.680","Text":"In fact, you might ask why stop at the positive integers and 0,"},{"Start":"01:42.680 ","End":"01:44.705","Text":"what about negative integers?"},{"Start":"01:44.705 ","End":"01:48.125","Text":"Negative integers takes us right into rule number 7."},{"Start":"01:48.125 ","End":"01:57.990","Text":"Rule number 7 says that a^ minus n is 1 over a^n."},{"Start":"01:57.990 ","End":"01:59.010","Text":"That looks a bit strange,"},{"Start":"01:59.010 ","End":"02:00.590","Text":"there is nothing negative here."},{"Start":"02:00.590 ","End":"02:02.600","Text":"Here we have a negative number,"},{"Start":"02:02.600 ","End":"02:08.270","Text":"but negative becomes division when we talk about multiplication."},{"Start":"02:08.270 ","End":"02:09.710","Text":"When talking about plus and minus,"},{"Start":"02:09.710 ","End":"02:11.600","Text":"the opposite of plus is minus,"},{"Start":"02:11.600 ","End":"02:14.645","Text":"with multiplication, the opposite is division."},{"Start":"02:14.645 ","End":"02:18.545","Text":"Anyway, let me give you an example of how we use this."},{"Start":"02:18.545 ","End":"02:23.960","Text":"If I apply this rule to the example 3^minus 2,"},{"Start":"02:23.960 ","End":"02:27.910","Text":"this would tell me what 3^ minus 2 is."},{"Start":"02:27.910 ","End":"02:33.010","Text":"This would equal 1 over 3^2."},{"Start":"02:33.010 ","End":"02:34.680","Text":"If n is 2,"},{"Start":"02:34.680 ","End":"02:36.235","Text":"then this is minus 2."},{"Start":"02:36.235 ","End":"02:40.010","Text":"I just put it on the denominator and throughout the minus."},{"Start":"02:40.010 ","End":"02:42.890","Text":"In fact, this equals 1 over 9, but as I say,"},{"Start":"02:42.890 ","End":"02:45.650","Text":"less important as to the computation."},{"Start":"02:45.650 ","End":"02:49.425","Text":"By the way, this is also a definition as well as a rule."},{"Start":"02:49.425 ","End":"02:52.065","Text":"I\u0027d like to explain why this is so."},{"Start":"02:52.065 ","End":"02:55.100","Text":"Once again, just like in rule number 6,"},{"Start":"02:55.100 ","End":"02:57.980","Text":"we want to make sure that all the rules still apply."},{"Start":"02:57.980 ","End":"03:00.710","Text":"In fact, I could even take this rule again,"},{"Start":"03:00.710 ","End":"03:02.755","Text":"this a^m minus n,"},{"Start":"03:02.755 ","End":"03:08.100","Text":"what would happen if I put 3^0 minus 2?"},{"Start":"03:08.100 ","End":"03:10.515","Text":"I take m is 0 and n is 2."},{"Start":"03:10.515 ","End":"03:16.560","Text":"This would equal 3^0 over 3^2 according to this rule."},{"Start":"03:16.560 ","End":"03:20.805","Text":"But 3^0 according to Rule 6 is 1."},{"Start":"03:20.805 ","End":"03:24.300","Text":"So it\u0027s 1 over 3^2 just as here."},{"Start":"03:24.300 ","End":"03:26.910","Text":"That\u0027s why this rule makes sense,"},{"Start":"03:26.910 ","End":"03:29.395","Text":"a^minus n is 1 over a^n."},{"Start":"03:29.395 ","End":"03:31.850","Text":"Now there\u0027s a couple of special cases."},{"Start":"03:31.850 ","End":"03:36.200","Text":"In fact, there\u0027s one special case here that I\u0027d like to write separately."},{"Start":"03:36.200 ","End":"03:39.845","Text":"This is what happens if n=1."},{"Start":"03:39.845 ","End":"03:43.850","Text":"If I take this rule and put n=1,"},{"Start":"03:43.850 ","End":"03:51.395","Text":"it tells me that a^minus1 is 1 over a^1."},{"Start":"03:51.395 ","End":"03:54.660","Text":"Now, a^1, for example,"},{"Start":"03:54.660 ","End":"03:56.760","Text":"if I had 3^1,"},{"Start":"03:56.760 ","End":"03:59.850","Text":"it just means multiplying 1 factor of 3,"},{"Start":"03:59.850 ","End":"04:06.405","Text":"so a^1 would just be a. I can erase this 1."},{"Start":"04:06.405 ","End":"04:13.310","Text":"In fact, that a^1 equals a is so important that I\u0027m thinking of giving it a half rule,"},{"Start":"04:13.310 ","End":"04:14.974","Text":"let\u0027s say Rule 6.5. Why not 6.5?"},{"Start":"04:14.974 ","End":"04:17.067","Text":"It\u0027s still not quite a rule,"},{"Start":"04:17.067 ","End":"04:21.725","Text":"but it should be a rule that a^1 is equal to a,"},{"Start":"04:21.725 ","End":"04:24.725","Text":"obvious but still worth stating."},{"Start":"04:24.725 ","End":"04:28.775","Text":"That\u0027s what gave me the possibility of dropping the 1 from here."},{"Start":"04:28.775 ","End":"04:33.320","Text":"This is such a special case of Rule 7 that I put it together on the same line with"},{"Start":"04:33.320 ","End":"04:38.555","Text":"the rule that something to the power of minus 1 means the reciprocal."},{"Start":"04:38.555 ","End":"04:40.460","Text":"There\u0027s no negatives here."},{"Start":"04:40.460 ","End":"04:43.285","Text":"It\u0027s just 1 over."},{"Start":"04:43.285 ","End":"04:48.410","Text":"Rule number 8 is less often used but still important and"},{"Start":"04:48.410 ","End":"04:53.030","Text":"useful and basically continuous rule number 7 to the case of fractions."},{"Start":"04:53.030 ","End":"04:59.600","Text":"If I have a fraction and I raise this fraction to a power of a negative number,"},{"Start":"04:59.600 ","End":"05:03.800","Text":"what this rule says is I can take the fraction upside down,"},{"Start":"05:03.800 ","End":"05:08.600","Text":"the reciprocal and make the exponent positive."},{"Start":"05:08.600 ","End":"05:11.315","Text":"Let me explain why this is true."},{"Start":"05:11.315 ","End":"05:14.150","Text":"I\u0027ll use a numerical example to explain,"},{"Start":"05:14.150 ","End":"05:15.770","Text":"although it\u0027s pretty general."},{"Start":"05:15.770 ","End":"05:20.360","Text":"Let\u0027s say I have 3 over 5^2,"},{"Start":"05:20.360 ","End":"05:23.285","Text":"on the one hand,"},{"Start":"05:23.285 ","End":"05:26.430","Text":"if I use rule number 5,"},{"Start":"05:26.430 ","End":"05:27.825","Text":"I believe it was,"},{"Start":"05:27.825 ","End":"05:30.500","Text":"it says we can take the exponent in"},{"Start":"05:30.500 ","End":"05:35.180","Text":"the numerator separately and then the denominator separately."},{"Start":"05:35.180 ","End":"05:42.790","Text":"Let\u0027s take (3 over 5)^minus 2."},{"Start":"05:42.790 ","End":"05:45.830","Text":"Now according to rule number 7,"},{"Start":"05:45.830 ","End":"05:55.550","Text":"something to the power of minus 2 would be 1 over the same thing to the power of plus 2."},{"Start":"05:55.550 ","End":"06:00.130","Text":"Now continuing, this is equal to 1 over,"},{"Start":"06:00.130 ","End":"06:02.930","Text":"when I take something to the power of 2,"},{"Start":"06:02.930 ","End":"06:08.900","Text":"then I take the numerator separately and the denominator separately."},{"Start":"06:08.900 ","End":"06:12.160","Text":"I think this was rule number 5."},{"Start":"06:12.160 ","End":"06:15.690","Text":"Now using properties of fractions,"},{"Start":"06:15.690 ","End":"06:19.820","Text":"1 over a fraction is the reciprocal of a fraction,"},{"Start":"06:19.820 ","End":"06:21.350","Text":"the upside down fraction."},{"Start":"06:21.350 ","End":"06:23.840","Text":"This is a property of fractions in general."},{"Start":"06:23.840 ","End":"06:25.945","Text":"When I take 1 over a fraction,"},{"Start":"06:25.945 ","End":"06:28.020","Text":"I invert the fraction."},{"Start":"06:28.020 ","End":"06:31.655","Text":"Now 5^2 over 3^2,"},{"Start":"06:31.655 ","End":"06:34.505","Text":"I just use the same one,"},{"Start":"06:34.505 ","End":"06:43.145","Text":"I mean number 5 backwards and say that this is equal to 5 over 3^2."},{"Start":"06:43.145 ","End":"06:44.960","Text":"The rule reads from right to left."},{"Start":"06:44.960 ","End":"06:48.965","Text":"We\u0027re taking it from left to right because equality works both ways."},{"Start":"06:48.965 ","End":"06:54.990","Text":"We see that this shows that Rule 8 does work and why it works."},{"Start":"06:54.990 ","End":"06:57.210","Text":"I had (3 over 2)^minus 2,"},{"Start":"06:57.210 ","End":"07:02.020","Text":"and then it comes out as (5 over 3)^plus 2."},{"Start":"07:02.020 ","End":"07:04.580","Text":"It\u0027s basically because of Rule 7,"},{"Start":"07:04.580 ","End":"07:06.935","Text":"which allows us to put this in the denominator."},{"Start":"07:06.935 ","End":"07:10.475","Text":"Because of the properties of fractions that says when we divide by a fraction,"},{"Start":"07:10.475 ","End":"07:13.110","Text":"we take the reciprocal."}],"ID":8102},{"Watched":false,"Name":"Rules 9-10","Duration":"8m 9s","ChapterTopicVideoID":8010,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8010.jpeg","UploadDate":"2020-09-30T14:42:03.2030000","DurationForVideoObject":"PT8M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"Now we come to rule number 9,"},{"Start":"00:02.730 ","End":"00:05.265","Text":"which is made up of 3 parts."},{"Start":"00:05.265 ","End":"00:07.590","Text":"Why don\u0027t I take them one at a time?"},{"Start":"00:07.590 ","End":"00:10.620","Text":"This is the first part of Rule 9."},{"Start":"00:10.620 ","End":"00:15.210","Text":"The square root of a is a^1/2,"},{"Start":"00:15.210 ","End":"00:18.654","Text":"or sometimes you read it the other way,"},{"Start":"00:18.654 ","End":"00:20.040","Text":"equals works both ways."},{"Start":"00:20.040 ","End":"00:23.180","Text":"Actually, I find it more commonly to say something to"},{"Start":"00:23.180 ","End":"00:26.450","Text":"the power of a 1/2 is the square root of that something,"},{"Start":"00:26.450 ","End":"00:29.320","Text":"and I\u0027ll give an example."},{"Start":"00:29.320 ","End":"00:31.750","Text":"Say, 9^1/2,"},{"Start":"00:31.970 ","End":"00:33.980","Text":"according to this rule,"},{"Start":"00:33.980 ","End":"00:38.765","Text":"it\u0027s just equal to the square root of 9, which is 3."},{"Start":"00:38.765 ","End":"00:41.270","Text":"Perhaps at the end, I\u0027ll give you the logic of"},{"Start":"00:41.270 ","End":"00:44.255","Text":"why something to the power of a 1/2 is the square root."},{"Start":"00:44.255 ","End":"00:46.390","Text":"Meanwhile, just accept it as a rule."},{"Start":"00:46.390 ","End":"00:49.115","Text":"Then I want to remind you something I said before,"},{"Start":"00:49.115 ","End":"00:52.640","Text":"that exponents and roots are very closely related,"},{"Start":"00:52.640 ","End":"00:56.365","Text":"so it\u0027s rules of exponents and roots."},{"Start":"00:56.365 ","End":"01:00.990","Text":"All 3 parts of Rule 9 all relate to roots."},{"Start":"01:00.990 ","End":"01:03.740","Text":"The next part is similar to this."},{"Start":"01:03.740 ","End":"01:10.430","Text":"It\u0027s actually a generalization that the nth root of a is a^1."},{"Start":"01:10.430 ","End":"01:14.145","Text":"In this rule, n can be 2,"},{"Start":"01:14.145 ","End":"01:15.690","Text":"3, 4,"},{"Start":"01:15.690 ","End":"01:17.105","Text":"5, and so on."},{"Start":"01:17.105 ","End":"01:19.895","Text":"We don\u0027t usually take the oneth root."},{"Start":"01:19.895 ","End":"01:21.275","Text":"Theoretically, you could,"},{"Start":"01:21.275 ","End":"01:23.810","Text":"you could say the oneth root of a is just a,"},{"Start":"01:23.810 ","End":"01:25.550","Text":"but it\u0027s not usually done."},{"Start":"01:25.550 ","End":"01:28.135","Text":"We start from 2 onwards."},{"Start":"01:28.135 ","End":"01:30.045","Text":"I\u0027ll give you an example."},{"Start":"01:30.045 ","End":"01:33.855","Text":"If I asked you what is a^1/3,"},{"Start":"01:33.855 ","End":"01:38.505","Text":"that would be the cube root of 8,"},{"Start":"01:38.505 ","End":"01:40.740","Text":"and that is equal to 2."},{"Start":"01:40.740 ","End":"01:46.270","Text":"Another example, let\u0027s say 32^1/5,"},{"Start":"01:47.030 ","End":"01:51.810","Text":"that is the fifth root of 32."},{"Start":"01:51.810 ","End":"01:56.300","Text":"I\u0027m mentally asking myself what to the power of 5 is 32 and you know this,"},{"Start":"01:56.300 ","End":"02:00.275","Text":"it\u0027s 2 because 2^5 is 32."},{"Start":"02:00.275 ","End":"02:02.090","Text":"Notice that, as I said,"},{"Start":"02:02.090 ","End":"02:06.590","Text":"this is a special case of this because when I write square root without anything,"},{"Start":"02:06.590 ","End":"02:09.020","Text":"it\u0027s understood to be the second root."},{"Start":"02:09.020 ","End":"02:12.790","Text":"This 2 is optional and usually omitted."},{"Start":"02:12.790 ","End":"02:15.060","Text":"If you put n equals 2 here,"},{"Start":"02:15.060 ","End":"02:18.930","Text":"you get a^1/2 is the second root of a,"},{"Start":"02:18.930 ","End":"02:21.330","Text":"which is just the regular square root."},{"Start":"02:21.330 ","End":"02:23.960","Text":"You don\u0027t have to worry about square root of"},{"Start":"02:23.960 ","End":"02:29.510","Text":"negative numbers because we said that whenever we take a^n,"},{"Start":"02:29.510 ","End":"02:36.535","Text":"by definition, we restrict a to being positive when using rules of exponents."},{"Start":"02:36.535 ","End":"02:40.525","Text":"Now let\u0027s get to the third part of Rule 9."},{"Start":"02:40.525 ","End":"02:43.955","Text":"More frequently, I read the equals from right to left."},{"Start":"02:43.955 ","End":"02:45.730","Text":"Of course, equals works both ways."},{"Start":"02:45.730 ","End":"02:49.010","Text":"Really everything should be with a double arrow."},{"Start":"02:49.010 ","End":"02:50.480","Text":"But more frequently,"},{"Start":"02:50.480 ","End":"02:51.710","Text":"I go from right to left."},{"Start":"02:51.710 ","End":"02:54.530","Text":"Let me give an example of this."},{"Start":"02:54.530 ","End":"02:58.415","Text":"If I have to compute 8^2/3,"},{"Start":"02:58.415 ","End":"03:02.240","Text":"then according to this rule, from right to left,"},{"Start":"03:02.240 ","End":"03:06.985","Text":"it\u0027s the cube root of 8^2,"},{"Start":"03:06.985 ","End":"03:12.195","Text":"which is the cube root of 64,"},{"Start":"03:12.195 ","End":"03:15.690","Text":"and the cube root of 64 is 4,"},{"Start":"03:15.690 ","End":"03:18.290","Text":"because 4 times 4 times 4 is 64."},{"Start":"03:18.290 ","End":"03:20.630","Text":"Also here, let\u0027s take an example,"},{"Start":"03:20.630 ","End":"03:24.635","Text":"32, I\u0027ll also change the 1 to a 2,"},{"Start":"03:24.635 ","End":"03:29.720","Text":"32^2/5 is the fifth root of 32^2,"},{"Start":"03:29.720 ","End":"03:35.855","Text":"which is the fifth root of 1,024."},{"Start":"03:35.855 ","End":"03:40.370","Text":"I\u0027ll write it fifth root of 1,024."},{"Start":"03:40.370 ","End":"03:43.575","Text":"This happens to be 4."},{"Start":"03:43.575 ","End":"03:49.045","Text":"You can check 4 times 4 times 4 times 4 times 4 is 1,024."},{"Start":"03:49.045 ","End":"03:51.410","Text":"Now this sometimes gets into large numbers,"},{"Start":"03:51.410 ","End":"03:52.940","Text":"if I had 32 cubed."},{"Start":"03:52.940 ","End":"03:58.615","Text":"Actually, there\u0027s a variation of this rule which is sometimes more helpful,"},{"Start":"03:58.615 ","End":"04:02.435","Text":"and this says that a^m is also equal to,"},{"Start":"04:02.435 ","End":"04:05.510","Text":"instead of raising to the power of m and taking the nth root,"},{"Start":"04:05.510 ","End":"04:07.205","Text":"I can do it the other way round."},{"Start":"04:07.205 ","End":"04:11.450","Text":"I can take the nth root of a and then raise"},{"Start":"04:11.450 ","End":"04:16.940","Text":"that result to the power of m. For example here,"},{"Start":"04:16.940 ","End":"04:19.460","Text":"I could have done the computation differently."},{"Start":"04:19.460 ","End":"04:23.210","Text":"I could\u0027ve said that this is equal to, first of all,"},{"Start":"04:23.210 ","End":"04:27.765","Text":"the cube root of 8 and then to the power of 2."},{"Start":"04:27.765 ","End":"04:29.640","Text":"The cube root of 8 is 2,"},{"Start":"04:29.640 ","End":"04:31.660","Text":"and 2^2 is 4."},{"Start":"04:31.660 ","End":"04:33.185","Text":"Of course, you get the same answer."},{"Start":"04:33.185 ","End":"04:37.415","Text":"Unlike here, I could have said that it\u0027s equal to"},{"Start":"04:37.415 ","End":"04:43.580","Text":"the fifth root of 32 and then raised to the power of 2,"},{"Start":"04:43.580 ","End":"04:45.260","Text":"which is fifth root of 32,"},{"Start":"04:45.260 ","End":"04:50.865","Text":"where we have it here is 2^2, which is 4."},{"Start":"04:50.865 ","End":"04:56.989","Text":"This is often more useful computationally because this tends to get large numbers."},{"Start":"04:56.989 ","End":"04:59.420","Text":"We have 32^4."},{"Start":"04:59.420 ","End":"05:01.190","Text":"I just wanted to point out, of course,"},{"Start":"05:01.190 ","End":"05:03.425","Text":"that this is a generalization of this."},{"Start":"05:03.425 ","End":"05:07.475","Text":"That if I put m equals 1 here,"},{"Start":"05:07.475 ","End":"05:08.780","Text":"if we put m equals 1,"},{"Start":"05:08.780 ","End":"05:12.200","Text":"and a^1 is just a and I get this rule."},{"Start":"05:12.200 ","End":"05:14.630","Text":"Each one is a generalization of the previous."},{"Start":"05:14.630 ","End":"05:17.083","Text":"This generalizes this, which generalizes this,"},{"Start":"05:17.083 ","End":"05:20.735","Text":"and this is the most general with the fractional power."},{"Start":"05:20.735 ","End":"05:22.970","Text":"Now we come to Rule 10,"},{"Start":"05:22.970 ","End":"05:24.830","Text":"which is very short."},{"Start":"05:24.830 ","End":"05:30.770","Text":"It just says that a^n is always positive."},{"Start":"05:30.770 ","End":"05:34.340","Text":"This is in the context of rules of exponents."},{"Start":"05:34.340 ","End":"05:37.100","Text":"Like we said before, in this context,"},{"Start":"05:37.100 ","End":"05:41.515","Text":"we\u0027re always assuming that a is positive."},{"Start":"05:41.515 ","End":"05:44.690","Text":"I\u0027ll just divide it up into 3 cases here."},{"Start":"05:44.690 ","End":"05:47.945","Text":"In case 1, I\u0027ll take n as positive,"},{"Start":"05:47.945 ","End":"05:50.795","Text":"second case n is 0,"},{"Start":"05:50.795 ","End":"05:53.615","Text":"and third case, n is negative."},{"Start":"05:53.615 ","End":"05:55.490","Text":"Well, if this is the case,"},{"Start":"05:55.490 ","End":"05:57.650","Text":"then if n is bigger than 0,"},{"Start":"05:57.650 ","End":"06:03.440","Text":"then I have certainly positive to the power of positive will be positive."},{"Start":"06:03.440 ","End":"06:06.200","Text":"There is nowhere for a negative to creep in here."},{"Start":"06:06.200 ","End":"06:10.315","Text":"If n is 0, then I have a^0,"},{"Start":"06:10.315 ","End":"06:11.460","Text":"which is 1,"},{"Start":"06:11.460 ","End":"06:13.665","Text":"which is certainly positive."},{"Start":"06:13.665 ","End":"06:15.650","Text":"If n is negative,"},{"Start":"06:15.650 ","End":"06:22.100","Text":"then let\u0027s say that n is equal to minus, say m,"},{"Start":"06:22.100 ","End":"06:23.915","Text":"where m is positive,"},{"Start":"06:23.915 ","End":"06:27.115","Text":"so I have a^minus m,"},{"Start":"06:27.115 ","End":"06:33.600","Text":"and this is equal to by one of the previous rules, 1/a^m."},{"Start":"06:34.370 ","End":"06:37.250","Text":"If this is positive,"},{"Start":"06:37.250 ","End":"06:39.935","Text":"then we get 1/positive,"},{"Start":"06:39.935 ","End":"06:41.960","Text":"which is also positive,"},{"Start":"06:41.960 ","End":"06:44.479","Text":"meaning bigger than 0."},{"Start":"06:44.479 ","End":"06:46.415","Text":"That\u0027s this rule."},{"Start":"06:46.415 ","End":"06:50.330","Text":"Now we\u0027re done with this clip except if you want to stay and I want to give you"},{"Start":"06:50.330 ","End":"06:54.380","Text":"an explanation as to why this is true."},{"Start":"06:54.380 ","End":"06:55.670","Text":"It\u0027s not mandatory,"},{"Start":"06:55.670 ","End":"06:58.520","Text":"so feel free to skip the rest."},{"Start":"06:58.520 ","End":"07:00.784","Text":"Anyway, for those who are interested,"},{"Start":"07:00.784 ","End":"07:02.395","Text":"why is this true?"},{"Start":"07:02.395 ","End":"07:05.010","Text":"Why is a^1/2,"},{"Start":"07:05.010 ","End":"07:08.400","Text":"the square root of a doesn\u0027t seem intuitive?"},{"Start":"07:08.400 ","End":"07:10.190","Text":"Let me explain it this way."},{"Start":"07:10.190 ","End":"07:12.800","Text":"We want the rules of exponents to hold."},{"Start":"07:12.800 ","End":"07:14.365","Text":"We say a^1/2,"},{"Start":"07:14.365 ","End":"07:15.818","Text":"let\u0027s see what it could be."},{"Start":"07:15.818 ","End":"07:17.945","Text":"Let\u0027s call it x for the moment."},{"Start":"07:17.945 ","End":"07:21.740","Text":"Now if I raise both sides to the power of 2,"},{"Start":"07:21.740 ","End":"07:27.455","Text":"I\u0027ll get (a^1/2)^2 equals x squared."},{"Start":"07:27.455 ","End":"07:29.000","Text":"I forget the numbers of the rules,"},{"Start":"07:29.000 ","End":"07:31.385","Text":"but there\u0027s a rule that the power of a power,"},{"Start":"07:31.385 ","End":"07:34.025","Text":"in that situation, you multiply the powers."},{"Start":"07:34.025 ","End":"07:39.620","Text":"We get a^1/2 times 2 is x squared,"},{"Start":"07:39.620 ","End":"07:43.280","Text":"1/2 times 2 is 1, a^1 is a."},{"Start":"07:43.280 ","End":"07:48.735","Text":"We get that a is equal to x^2,"},{"Start":"07:48.735 ","End":"07:50.353","Text":"x is going to be positive,"},{"Start":"07:50.353 ","End":"07:52.070","Text":"because we said, here,"},{"Start":"07:52.070 ","End":"07:54.755","Text":"a^n is always positive,"},{"Start":"07:54.755 ","End":"07:56.375","Text":"so x is positive,"},{"Start":"07:56.375 ","End":"08:01.250","Text":"so x is going to equal the square root of a."},{"Start":"08:01.250 ","End":"08:03.755","Text":"If x^2 is a and x is positive,"},{"Start":"08:03.755 ","End":"08:09.900","Text":"then x is square root of a and that explains this rule. Now we\u0027re done."}],"ID":8103},{"Watched":false,"Name":"Rules 11-12","Duration":"2m 41s","ChapterTopicVideoID":8011,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8011.jpeg","UploadDate":"2020-09-30T13:49:40.2800000","DurationForVideoObject":"PT2M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:03.900","Text":"Now we come to rules 11 and 12,"},{"Start":"00:03.900 ","End":"00:05.955","Text":"and these should be the last."},{"Start":"00:05.955 ","End":"00:09.690","Text":"If I wanted to take the nth root of a product A times B,"},{"Start":"00:09.690 ","End":"00:13.980","Text":"I can take the nth root of B separately and multiply it by the nth root of B."},{"Start":"00:13.980 ","End":"00:18.060","Text":"Similarly, the multiplication works with division the nth root of"},{"Start":"00:18.060 ","End":"00:22.395","Text":"this over this is the nth root of this over the nth root of that."},{"Start":"00:22.395 ","End":"00:24.870","Text":"I\u0027ll give you an example of each."},{"Start":"00:24.870 ","End":"00:35.310","Text":"If I want let\u0027s say the 4th root of 625 times 16,"},{"Start":"00:35.310 ","End":"00:39.300","Text":"then I can say that this is equal to the 4th root of"},{"Start":"00:39.300 ","End":"00:46.530","Text":"625 separately times the 4th root of 16."},{"Start":"00:46.530 ","End":"00:49.620","Text":"This is equal to 5."},{"Start":"00:49.620 ","End":"00:53.445","Text":"You can check 5 times 5 times 5 times 5 is 625."},{"Start":"00:53.445 ","End":"00:56.160","Text":"The fourth root of 16, we know is 2,"},{"Start":"00:56.160 ","End":"00:58.245","Text":"2 to the 4th is 16."},{"Start":"00:58.245 ","End":"01:00.885","Text":"The answer is 10."},{"Start":"01:00.885 ","End":"01:05.705","Text":"Of course we could have multiplied out first and said that the 4th root of"},{"Start":"01:05.705 ","End":"01:10.530","Text":"this times this is 10,000 and 10,000 is 1 with 4 zeros."},{"Start":"01:10.530 ","End":"01:12.060","Text":"Of course the 4th root is 10,"},{"Start":"01:12.060 ","End":"01:14.410","Text":"so it works the other way too."},{"Start":"01:14.410 ","End":"01:16.625","Text":"Now, an example of this."},{"Start":"01:16.625 ","End":"01:19.140","Text":"If I want to know, let\u0027s say n is 3,"},{"Start":"01:19.140 ","End":"01:23.300","Text":"the cube root of 8/27."},{"Start":"01:23.300 ","End":"01:31.715","Text":"I can say that this is the cube root of 8 separately over the cube root of 27."},{"Start":"01:31.715 ","End":"01:33.410","Text":"This of course is 2,"},{"Start":"01:33.410 ","End":"01:36.830","Text":"this is 3, so the answer is 2/3."},{"Start":"01:36.830 ","End":"01:40.490","Text":"Now, these rules are more commonly given in"},{"Start":"01:40.490 ","End":"01:45.650","Text":"a special case where n is 2 and we take square roots, just copy-pasted here."},{"Start":"01:45.650 ","End":"01:47.795","Text":"If I replace this by square roots,"},{"Start":"01:47.795 ","End":"01:48.980","Text":"I don\u0027t have to write the 2,"},{"Start":"01:48.980 ","End":"01:50.840","Text":"I don\u0027t write anything."},{"Start":"01:50.840 ","End":"01:52.010","Text":"It means square root,"},{"Start":"01:52.010 ","End":"01:56.480","Text":"so I\u0027m just erasing the n. This is how it\u0027s normally given."},{"Start":"01:56.480 ","End":"01:58.100","Text":"Is it the same rule?"},{"Start":"01:58.100 ","End":"02:01.520","Text":"Well, maybe I\u0027ll call this 11a and12a."},{"Start":"02:01.520 ","End":"02:04.160","Text":"Anyway, same thing just with n is 2."},{"Start":"02:04.160 ","End":"02:05.705","Text":"We\u0027re done with this,"},{"Start":"02:05.705 ","End":"02:10.020","Text":"except that I would like to show you where these 2 rules come from."},{"Start":"02:10.550 ","End":"02:15.275","Text":"I\u0027m going to remind you that rule 4 rule 5 where this,"},{"Start":"02:15.275 ","End":"02:19.190","Text":"and it\u0027s very similar to this because if I replace in each of these rules,"},{"Start":"02:19.190 ","End":"02:23.000","Text":"if I replace n by 1,"},{"Start":"02:23.000 ","End":"02:25.055","Text":"raise everything to the power of n,"},{"Start":"02:25.055 ","End":"02:26.405","Text":"I get the nth root,"},{"Start":"02:26.405 ","End":"02:29.345","Text":"cause to the power of 1 is like the nth root."},{"Start":"02:29.345 ","End":"02:31.130","Text":"If I replace this here,"},{"Start":"02:31.130 ","End":"02:33.665","Text":"then these 2 rules will give me these 2 rules."},{"Start":"02:33.665 ","End":"02:38.060","Text":"That\u0027s the rationale, that\u0027s the proof where they come from."},{"Start":"02:38.060 ","End":"02:41.969","Text":"That\u0027s it for these rules."}],"ID":8104},{"Watched":false,"Name":"Afterword","Duration":"3m 25s","ChapterTopicVideoID":13797,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/13797.jpeg","UploadDate":"2020-09-30T13:53:17.9170000","DurationForVideoObject":"PT3M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.590","Text":"This is the optional section where I explain why in the rules of exponents,"},{"Start":"00:07.590 ","End":"00:11.189","Text":"why do we restrict the base of an exponent to be positive?"},{"Start":"00:11.189 ","End":"00:14.909","Text":"In other words, when I take a to the power of n,"},{"Start":"00:14.909 ","End":"00:16.590","Text":"That this part a,"},{"Start":"00:16.590 ","End":"00:19.440","Text":"the base we take to be positive."},{"Start":"00:19.440 ","End":"00:20.790","Text":"There\u0027s quite a few reasons,"},{"Start":"00:20.790 ","End":"00:22.320","Text":"quite a few problems that happen."},{"Start":"00:22.320 ","End":"00:24.525","Text":"I\u0027ll just illustrate a couple of problems."},{"Start":"00:24.525 ","End":"00:27.360","Text":"But in fact, all of the problems stem from rule number"},{"Start":"00:27.360 ","End":"00:30.210","Text":"9 and from fractional exponents and roots."},{"Start":"00:30.210 ","End":"00:33.750","Text":"That if we didn\u0027t have fractional exponents and roots,"},{"Start":"00:33.750 ","End":"00:38.730","Text":"there might not be a good reason and might all work out."},{"Start":"00:38.730 ","End":"00:44.690","Text":"But certainly with this rule and with fractional exponents and roots, things go wrong."},{"Start":"00:44.690 ","End":"00:46.595","Text":"I\u0027ll show you, for example,"},{"Start":"00:46.595 ","End":"00:48.560","Text":"at the most basic level,"},{"Start":"00:48.560 ","End":"00:50.645","Text":"certain things aren\u0027t defined."},{"Start":"00:50.645 ","End":"00:53.900","Text":"For example, minus 4 to the power of a"},{"Start":"00:53.900 ","End":"00:58.460","Text":"0.5 would not be defined because according to this rule,"},{"Start":"00:58.460 ","End":"01:02.105","Text":"this would be the square root of minus 4."},{"Start":"01:02.105 ","End":"01:03.890","Text":"The square root of minus 4,"},{"Start":"01:03.890 ","End":"01:08.180","Text":"we know does not exist because no number squared is a negative number."},{"Start":"01:08.180 ","End":"01:16.147","Text":"But it\u0027s even worse because even if I take something like minus 4 to the power of 3."},{"Start":"01:16.147 ","End":"01:19.909","Text":"Which at first sight is defined,"},{"Start":"01:19.909 ","End":"01:21.020","Text":"you could say, well,"},{"Start":"01:21.020 ","End":"01:23.900","Text":"I could take minus 4 times minus 4 times minus 4."},{"Start":"01:23.900 ","End":"01:27.725","Text":"But if mathematics is to be consistent,"},{"Start":"01:27.725 ","End":"01:34.345","Text":"surely I could then be permitted to write it as minus 4 to the power of 6 over 2,"},{"Start":"01:34.345 ","End":"01:37.550","Text":"because 3 is 6 over 2 after all."},{"Start":"01:37.550 ","End":"01:41.825","Text":"But then I would have to allow this rule."},{"Start":"01:41.825 ","End":"01:44.555","Text":"This rule, if it\u0027s going to hold in general."},{"Start":"01:44.555 ","End":"01:47.795","Text":"Then I would say that this equals,"},{"Start":"01:47.795 ","End":"01:51.830","Text":"by the right-hand side interpretation,"},{"Start":"01:51.830 ","End":"01:58.370","Text":"this would equal the square root of a second route,"},{"Start":"01:58.370 ","End":"02:00.470","Text":"but we don\u0027t have to write second root."},{"Start":"02:00.470 ","End":"02:05.540","Text":"I\u0027m going to erase. Then raised to the power of 6."},{"Start":"02:05.540 ","End":"02:06.920","Text":"Sorry, I said a,"},{"Start":"02:06.920 ","End":"02:09.455","Text":"but we have our a is minus 4."},{"Start":"02:09.455 ","End":"02:13.445","Text":"What we would get would be minus 4."},{"Start":"02:13.445 ","End":"02:15.620","Text":"I can use either this one or this one."},{"Start":"02:15.620 ","End":"02:19.995","Text":"Let me use this. I take the second root."},{"Start":"02:19.995 ","End":"02:23.330","Text":"Then to the power of m,"},{"Start":"02:23.330 ","End":"02:24.680","Text":"which is 6,"},{"Start":"02:24.680 ","End":"02:29.525","Text":"here I\u0027m taking this to be my m and this to be n. Now,"},{"Start":"02:29.525 ","End":"02:31.940","Text":"second root is just the same as square roots."},{"Start":"02:31.940 ","End":"02:34.160","Text":"I can erase. Once again,"},{"Start":"02:34.160 ","End":"02:37.040","Text":"I encountered the problem of the square root of a negative number."},{"Start":"02:37.040 ","End":"02:41.510","Text":"It\u0027s not the sixth is the problem is the square root of a minus 4 is a problem."},{"Start":"02:41.510 ","End":"02:43.980","Text":"Again, we run into trouble."},{"Start":"02:43.980 ","End":"02:47.195","Text":"I think there are other variations you could think of."},{"Start":"02:47.195 ","End":"02:53.810","Text":"But this should be enough to say that because of fractional exponents and roots,"},{"Start":"02:53.810 ","End":"02:56.420","Text":"that the rules break down,"},{"Start":"02:56.420 ","End":"02:58.475","Text":"other rules break down also."},{"Start":"02:58.475 ","End":"03:01.160","Text":"Rather than just say exceptionally that sometimes"},{"Start":"03:01.160 ","End":"03:03.710","Text":"the base can be positive and sometimes not."},{"Start":"03:03.710 ","End":"03:06.965","Text":"We just say that in rules of exponents,"},{"Start":"03:06.965 ","End":"03:08.930","Text":"the base should always be positive."},{"Start":"03:08.930 ","End":"03:12.740","Text":"It turns out it\u0027s not even a great restriction that it\u0027s very rare that we"},{"Start":"03:12.740 ","End":"03:17.470","Text":"need to use rules of exponents with negatives and we can\u0027t easily get around it."},{"Start":"03:17.470 ","End":"03:20.435","Text":"That\u0027s the reason for that."},{"Start":"03:20.435 ","End":"03:25.530","Text":"Now we really are done with this chapter on the rules of exponents."}],"ID":14582},{"Watched":false,"Name":"Exercise 1","Duration":"9m 2s","ChapterTopicVideoID":8021,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8021.jpeg","UploadDate":"2020-09-30T14:03:22.9400000","DurationForVideoObject":"PT9M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.920","Text":"In this exercise, there are 6 independent parts and each of them we have"},{"Start":"00:04.920 ","End":"00:10.200","Text":"to simplify and make sure that our answers don\u0027t contain negative exponents,"},{"Start":"00:10.200 ","End":"00:11.640","Text":"meaning only positive,"},{"Start":"00:11.640 ","End":"00:15.270","Text":"and in each of them there are many ways to solve it."},{"Start":"00:15.270 ","End":"00:16.875","Text":"There are many paths you can take."},{"Start":"00:16.875 ","End":"00:20.250","Text":"But I\u0027m going to assume that you\u0027re very familiar with all the rules of exponents."},{"Start":"00:20.250 ","End":"00:23.160","Text":"If not, you\u0027d better study them and then come back."},{"Start":"00:23.160 ","End":"00:25.545","Text":"Let\u0027s start with the first."},{"Start":"00:25.545 ","End":"00:28.320","Text":"I\u0027m going to suggest rewriting the order,"},{"Start":"00:28.320 ","End":"00:29.940","Text":"which is not something you have to do,"},{"Start":"00:29.940 ","End":"00:33.315","Text":"but it simplifies, makes it easier to see."},{"Start":"00:33.315 ","End":"00:37.845","Text":"If I write a^3a^4 times b^2b^5,"},{"Start":"00:37.845 ","End":"00:40.785","Text":"then I have a\u0027s together and the b\u0027s together."},{"Start":"00:40.785 ","End":"00:43.835","Text":"But you can totally skip this step if you want to."},{"Start":"00:43.835 ","End":"00:50.120","Text":"Because here we say a^3 times a^4 is a^3 plus 4,"},{"Start":"00:50.120 ","End":"00:52.250","Text":"which I right away is 7."},{"Start":"00:52.250 ","End":"00:56.630","Text":"Though you could have seen it from here that this times this is a^7."},{"Start":"00:56.630 ","End":"00:57.890","Text":"As you get more adept,"},{"Start":"00:57.890 ","End":"01:05.330","Text":"you take more shortcuts and then b^2 times b^5 is b^2 plus 5,"},{"Start":"01:05.330 ","End":"01:07.565","Text":"which is also b^7."},{"Start":"01:07.565 ","End":"01:13.175","Text":"I could\u0027ve taken another step and written 2 plus 5 here or 3 plus 4 here,"},{"Start":"01:13.175 ","End":"01:15.665","Text":"but I\u0027m assuming you have some familiarity,"},{"Start":"01:15.665 ","End":"01:17.375","Text":"and that\u0027s the answer."},{"Start":"01:17.375 ","End":"01:19.205","Text":"Onto the next one."},{"Start":"01:19.205 ","End":"01:24.485","Text":"There are, again, any number of paths we can do to solve this."},{"Start":"01:24.485 ","End":"01:32.045","Text":"One way of doing this is to write everything on the denominator as negative exponent."},{"Start":"01:32.045 ","End":"01:36.260","Text":"In general, you can take a factor across"},{"Start":"01:36.260 ","End":"01:38.660","Text":"the dividing line from denominator to"},{"Start":"01:38.660 ","End":"01:42.410","Text":"the numerator or the other way around and make its exponent negative."},{"Start":"01:42.410 ","End":"01:45.230","Text":"What I\u0027m going to do here is I\u0027m going to take everything to"},{"Start":"01:45.230 ","End":"01:50.120","Text":"the numerator and say it\u0027s x^7, y^5."},{"Start":"01:50.120 ","End":"01:54.485","Text":"Now y^2 on the numerator is y to the minus 2,"},{"Start":"01:54.485 ","End":"01:58.810","Text":"and x^3 on the numerator is x to the minus 3."},{"Start":"01:58.810 ","End":"02:02.780","Text":"Then I\u0027m not going to rearrange the order like I did before."},{"Start":"02:02.780 ","End":"02:06.545","Text":"It will take the x\u0027s separately and we see we have x^7,"},{"Start":"02:06.545 ","End":"02:08.195","Text":"x to the minus 3,"},{"Start":"02:08.195 ","End":"02:12.695","Text":"so we add the exponents this time I\u0027ll spell it out 7 minus 3,"},{"Start":"02:12.695 ","End":"02:18.650","Text":"the power of 5, y to the minus 2 is y to the power of 5 minus 2,"},{"Start":"02:18.650 ","End":"02:23.160","Text":"and this equals x^4 because 7 minus 3 is 4,"},{"Start":"02:23.160 ","End":"02:27.420","Text":"and 5 minus 2 is 3 and this is the answer."},{"Start":"02:27.420 ","End":"02:31.400","Text":"Okay. Next one. First thing I\u0027m going to do is use the rule for a power of"},{"Start":"02:31.400 ","End":"02:34.980","Text":"a power that\u0027s where you just multiply the exponents, the powers."},{"Start":"02:34.980 ","End":"02:41.060","Text":"The first thing I get is x^2 times 3 is 6, and the next one,"},{"Start":"02:41.060 ","End":"02:45.125","Text":"y to the power of 2 times 4 is 8,"},{"Start":"02:45.125 ","End":"02:48.475","Text":"and then x squared y cubed."},{"Start":"02:48.475 ","End":"02:50.210","Text":"Now I can rearrange the order,"},{"Start":"02:50.210 ","End":"02:58.055","Text":"but I\u0027m only going to do it mentally so I have x^6x^2 it\u0027s x^6 plus 2 is 8,"},{"Start":"02:58.055 ","End":"03:03.444","Text":"and y^8 plus 3 is 11."},{"Start":"03:03.444 ","End":"03:06.765","Text":"Scroll down a bit. Next one."},{"Start":"03:06.765 ","End":"03:10.880","Text":"Now I\u0027m going to use a couple of rules just to expand this."},{"Start":"03:10.880 ","End":"03:13.865","Text":"When you have a product that\u0027s raised to a power,"},{"Start":"03:13.865 ","End":"03:17.180","Text":"each of the factors in the product is raised to that power."},{"Start":"03:17.180 ","End":"03:24.050","Text":"When I cube this, I get 2^3 times x^3 times y^2,"},{"Start":"03:24.050 ","End":"03:25.745","Text":"all of this cubed,"},{"Start":"03:25.745 ","End":"03:27.230","Text":"I\u0027ll need the brackets here."},{"Start":"03:27.230 ","End":"03:30.980","Text":"Then the next one I have this times this to the power of 4,"},{"Start":"03:30.980 ","End":"03:35.025","Text":"so it\u0027s 2^4, x^4."},{"Start":"03:35.025 ","End":"03:37.685","Text":"Now I\u0027m going to rewrite this just,"},{"Start":"03:37.685 ","End":"03:42.620","Text":"I\u0027m going to calculate y^2^3."},{"Start":"03:42.620 ","End":"03:46.280","Text":"This bit is going to be y^6."},{"Start":"03:46.280 ","End":"03:48.860","Text":"Everything else is the same,"},{"Start":"03:48.860 ","End":"03:50.390","Text":"so I\u0027ll just copy it 2^3,"},{"Start":"03:50.390 ","End":"03:55.555","Text":"x^3, 2^4, x^4."},{"Start":"03:55.555 ","End":"03:58.410","Text":"Then I\u0027m going collect the terms together."},{"Start":"03:58.410 ","End":"04:00.090","Text":"The 2s together,"},{"Start":"04:00.090 ","End":"04:04.140","Text":"2^3 plus 4 is 2^7,"},{"Start":"04:04.140 ","End":"04:09.300","Text":"and then x^3 plus 4, x^7,"},{"Start":"04:09.300 ","End":"04:12.945","Text":"and y is only found here,"},{"Start":"04:12.945 ","End":"04:15.895","Text":"so that\u0027s all we have is y^6."},{"Start":"04:15.895 ","End":"04:18.470","Text":"There is one other thing that one could do,"},{"Start":"04:18.470 ","End":"04:20.240","Text":"and that is to evaluate,"},{"Start":"04:20.240 ","End":"04:21.380","Text":"since 2 is a number,"},{"Start":"04:21.380 ","End":"04:26.530","Text":"we can actually compute 2^7 and so we get that"},{"Start":"04:26.530 ","End":"04:32.570","Text":"this is 128x^7, y^6."},{"Start":"04:32.570 ","End":"04:36.200","Text":"Like I said, there are many paths to get to the answer each"},{"Start":"04:36.200 ","End":"04:39.800","Text":"time applying the rules necessarily in the same order."},{"Start":"04:39.800 ","End":"04:42.410","Text":"I\u0027m just giving you one suggested path to"},{"Start":"04:42.410 ","End":"04:45.885","Text":"the solution because the solution should all come out the same."},{"Start":"04:45.885 ","End":"04:48.810","Text":"Now, in number e,"},{"Start":"04:48.810 ","End":"04:55.490","Text":"what I\u0027d like to do first is raise these fractions to the exponent first of all,"},{"Start":"04:55.490 ","End":"05:02.015","Text":"so I get a^2^3 over b^3^3."},{"Start":"05:02.015 ","End":"05:05.690","Text":"This is using the rule that a fraction to a power,"},{"Start":"05:05.690 ","End":"05:09.655","Text":"I raise the numerator and denominator separately to that power,"},{"Start":"05:09.655 ","End":"05:11.970","Text":"and so similarly for the second thing,"},{"Start":"05:11.970 ","End":"05:17.600","Text":"I got b^2 over (a^2)^2."},{"Start":"05:17.600 ","End":"05:22.580","Text":"Now, I\u0027ll expand the power of a power by multiplying the exponents."},{"Start":"05:22.580 ","End":"05:25.970","Text":"So here I have a^2 times 3 is 6,"},{"Start":"05:25.970 ","End":"05:31.760","Text":"here I have b^3 times 3 is 9,"},{"Start":"05:32.730 ","End":"05:35.855","Text":"b^2, and a^4."},{"Start":"05:35.855 ","End":"05:39.380","Text":"Then I can proceed in any number of ways."},{"Start":"05:39.380 ","End":"05:43.625","Text":"What I might do is like in the earlier one,"},{"Start":"05:43.625 ","End":"05:48.770","Text":"to write the exponents in the denominator as negative exponent in the numerator."},{"Start":"05:48.770 ","End":"05:52.340","Text":"I\u0027ve got the first one a^6,"},{"Start":"05:52.340 ","End":"05:54.935","Text":"b to the minus 9,"},{"Start":"05:54.935 ","End":"05:57.220","Text":"and then b^2,"},{"Start":"05:57.220 ","End":"05:59.910","Text":"a to the minus 4."},{"Start":"05:59.910 ","End":"06:06.705","Text":"At this point I can collect the a\u0027s together and get a^6 minus 4,"},{"Start":"06:06.705 ","End":"06:09.030","Text":"if you like 6 plus minus 4,"},{"Start":"06:09.030 ","End":"06:11.835","Text":"still 2 so a^2,"},{"Start":"06:11.835 ","End":"06:17.695","Text":"and b to the power of minus 9 plus 2 is b to the power of minus 7."},{"Start":"06:17.695 ","End":"06:18.950","Text":"Now, while this is correct,"},{"Start":"06:18.950 ","End":"06:23.510","Text":"it doesn\u0027t answer the question which said no negative exponents allowed and so we"},{"Start":"06:23.510 ","End":"06:28.400","Text":"bring the negative exponent to the denominator and make it a positive exponent,"},{"Start":"06:28.400 ","End":"06:32.270","Text":"so it\u0027s a squared over b^7."},{"Start":"06:32.270 ","End":"06:36.755","Text":"Now we come to the last part of this question."},{"Start":"06:36.755 ","End":"06:38.690","Text":"So what I\u0027ll do,"},{"Start":"06:38.690 ","End":"06:42.170","Text":"I\u0027ll use the rule that a fraction to a power"},{"Start":"06:42.170 ","End":"06:46.520","Text":"means numerator and denominator separately to that power."},{"Start":"06:46.520 ","End":"06:50.510","Text":"So I have 4x^2,"},{"Start":"06:50.510 ","End":"06:53.040","Text":"y^3 to the power of 2,"},{"Start":"06:53.040 ","End":"06:55.650","Text":"and in the denominator,"},{"Start":"06:55.650 ","End":"06:59.205","Text":"x^4, y^5,"},{"Start":"06:59.205 ","End":"07:00.780","Text":"also to the power of 2."},{"Start":"07:00.780 ","End":"07:03.435","Text":"This same 2 here is what\u0027s here and here,"},{"Start":"07:03.435 ","End":"07:07.700","Text":"and now I can use the power of"},{"Start":"07:07.700 ","End":"07:14.030","Text":"a product to make each of these 3 parts that make this up."},{"Start":"07:14.030 ","End":"07:18.790","Text":"Raise each of these 3 parts to the power of 2 or square them so I get 4^2,"},{"Start":"07:18.790 ","End":"07:25.725","Text":"x^2^2, y^3^2, that\u0027s the numerator."},{"Start":"07:25.725 ","End":"07:29.310","Text":"Now the denominator made up of 2 bits, this and this,"},{"Start":"07:29.310 ","End":"07:35.220","Text":"each bit I square x^4^2,"},{"Start":"07:35.220 ","End":"07:38.440","Text":"y^5^2, and now let\u0027s see what we get."},{"Start":"07:38.930 ","End":"07:42.690","Text":"4^2 might as well evaluated as 16,"},{"Start":"07:42.690 ","End":"07:47.340","Text":"(x^2)^2 is x^2 times 2 is x^4,"},{"Start":"07:47.340 ","End":"07:54.015","Text":"and then y^3 times 2 is y^6, so numerator."},{"Start":"07:54.015 ","End":"08:02.565","Text":"Denominator I have x^4^2 is x^4 times 2 is 8,"},{"Start":"08:02.565 ","End":"08:07.820","Text":"y^5 times 2 is 10 and one possibility is to write"},{"Start":"08:07.820 ","End":"08:12.570","Text":"the exponent in the denominator as negative exponents in the numerator."},{"Start":"08:12.570 ","End":"08:16.385","Text":"I\u0027ll take the numerator as is because it\u0027s all positive there,"},{"Start":"08:16.385 ","End":"08:19.270","Text":"and then write x to the minus 8,"},{"Start":"08:19.270 ","End":"08:22.200","Text":"y to the minus 10,"},{"Start":"08:22.200 ","End":"08:25.220","Text":"and then let\u0027s just collect,"},{"Start":"08:25.220 ","End":"08:27.814","Text":"combine the same letters,"},{"Start":"08:27.814 ","End":"08:30.200","Text":"x\u0027s first and then y\u0027s."},{"Start":"08:30.200 ","End":"08:35.400","Text":"X we have to the power of 4 minus 8 is minus 4,"},{"Start":"08:35.500 ","End":"08:40.405","Text":"y^6 minus 10 is also minus 4."},{"Start":"08:40.405 ","End":"08:45.920","Text":"What I can do next is bring all the negative exponents to the denominator as"},{"Start":"08:45.920 ","End":"08:48.920","Text":"positive exponents so I\u0027m left with 16 for"},{"Start":"08:48.920 ","End":"08:52.640","Text":"a numerator and both of these go into the denominator."},{"Start":"08:52.640 ","End":"08:54.890","Text":"Then it\u0027s x^4,"},{"Start":"08:54.890 ","End":"08:58.260","Text":"y^4, and all the exponents are positive."},{"Start":"08:58.260 ","End":"09:02.490","Text":"That concludes Part F and the whole exercise."}],"ID":8114},{"Watched":false,"Name":"Exercise 2","Duration":"9m ","ChapterTopicVideoID":8013,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8013.jpeg","UploadDate":"2020-09-30T14:11:17.1300000","DurationForVideoObject":"PT9M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"This exercise is made up of 6 separate independent pieces,"},{"Start":"00:04.050 ","End":"00:06.240","Text":"and in each one we have to simplify"},{"Start":"00:06.240 ","End":"00:11.085","Text":"the expressions and get rid of any negative exponents."},{"Start":"00:11.085 ","End":"00:14.430","Text":"Let\u0027s start with the first, here it is."},{"Start":"00:14.430 ","End":"00:15.810","Text":"What I suggest doing,"},{"Start":"00:15.810 ","End":"00:19.110","Text":"is combining the a\u0027s first and then the b\u0027s."},{"Start":"00:19.110 ","End":"00:21.845","Text":"A^-3, a^4,"},{"Start":"00:21.845 ","End":"00:26.550","Text":"I use the property that when you multiply and you add the exponents,"},{"Start":"00:26.550 ","End":"00:30.850","Text":"so we have a^-3+4."},{"Start":"00:30.850 ","End":"00:37.290","Text":"Then this times this is b^-2+5."},{"Start":"00:37.290 ","End":"00:43.755","Text":"We get a^1 which is just a we\u0027ll write that in a moment,"},{"Start":"00:43.755 ","End":"00:49.065","Text":"and -2+3 is b^3 or b^3."},{"Start":"00:49.065 ","End":"00:56.585","Text":"Finally, just write it as ab^3 because we don\u0027t usually leave the power of 1."},{"Start":"00:56.585 ","End":"00:58.924","Text":"The next one I\u0027m going to use,"},{"Start":"00:58.924 ","End":"01:02.150","Text":"the rule of thumb derived from the rules,"},{"Start":"01:02.150 ","End":"01:06.770","Text":"but basically what it says is that you can switch sides from numerator to"},{"Start":"01:06.770 ","End":"01:11.660","Text":"denominator and vice versa and then you change the sign of the exponent."},{"Start":"01:11.660 ","End":"01:16.070","Text":"For example, x^7 is okay where it is,"},{"Start":"01:16.070 ","End":"01:17.315","Text":"it\u0027s a positive exponent,"},{"Start":"01:17.315 ","End":"01:25.910","Text":"y^5 goes to the denominator and then it becomes y^+5."},{"Start":"01:25.910 ","End":"01:28.775","Text":"Y^2 in the denominator is fine,"},{"Start":"01:28.775 ","End":"01:34.645","Text":"but the x^3 goes to the numerator and becomes x^+3."},{"Start":"01:34.645 ","End":"01:37.890","Text":"Essentially this goes down and this goes up,"},{"Start":"01:37.890 ","End":"01:45.060","Text":"and now we can combine x^7 times x^3 is x^7 plus 3 is x^10."},{"Start":"01:45.060 ","End":"01:47.250","Text":"Y^5, y^2,"},{"Start":"01:47.250 ","End":"01:54.005","Text":"is y^5+2 which is y^7 and that\u0027s it."},{"Start":"01:54.005 ","End":"01:55.535","Text":"In the next one,"},{"Start":"01:55.535 ","End":"01:58.160","Text":"we\u0027re going to have to first use the rule for power of"},{"Start":"01:58.160 ","End":"02:02.705","Text":"powers where we multiply the powers or exponents."},{"Start":"02:02.705 ","End":"02:07.730","Text":"We get x^-2 times 3 is -6,"},{"Start":"02:07.730 ","End":"02:12.610","Text":"y^-1 times 4 is -4,"},{"Start":"02:12.610 ","End":"02:15.642","Text":"and the rest of it as is for the moment."},{"Start":"02:15.642 ","End":"02:21.140","Text":"When I put all the ones with negative exponents into the denominator."},{"Start":"02:21.140 ","End":"02:24.290","Text":"Let\u0027s see. The only one that\u0027s positive is the y."},{"Start":"02:24.290 ","End":"02:27.440","Text":"So y^3 stays in the numerator,"},{"Start":"02:27.440 ","End":"02:30.230","Text":"all this is like a fraction over 1 originally."},{"Start":"02:30.230 ","End":"02:34.100","Text":"Anyway, I\u0027m pushing the first 3 into"},{"Start":"02:34.100 ","End":"02:40.955","Text":"the denominator as x^+6, y^+4, x^+2."},{"Start":"02:40.955 ","End":"02:44.539","Text":"Let\u0027s combine the x\u0027s and then combine the y\u0027s,"},{"Start":"02:44.539 ","End":"02:48.470","Text":"because I can change the order of multiplication anywhere I want. Let\u0027s see."},{"Start":"02:48.470 ","End":"02:53.670","Text":"X^-6 times x^-2 is x^-8."},{"Start":"02:53.670 ","End":"02:56.415","Text":"I just added negative 6 to negative 2."},{"Start":"02:56.415 ","End":"03:03.265","Text":"Here if I add negative 4 and 3, I get y^-1."},{"Start":"03:03.265 ","End":"03:08.329","Text":"When we have negative exponents in the numerator,"},{"Start":"03:08.329 ","End":"03:10.100","Text":"this is like a fraction over 1."},{"Start":"03:10.100 ","End":"03:11.600","Text":"I don\u0027t have to write this,"},{"Start":"03:11.600 ","End":"03:13.250","Text":"you imagine it that it\u0027s over 1."},{"Start":"03:13.250 ","End":"03:16.640","Text":"Then I put the negative ones in the denominator."},{"Start":"03:16.640 ","End":"03:18.650","Text":"Actually nothing is left in the numerator."},{"Start":"03:18.650 ","End":"03:26.010","Text":"When nothing\u0027s left, it\u0027s just 1 and then I have x^8 below and y^1 below,"},{"Start":"03:26.010 ","End":"03:27.750","Text":"which is just y,"},{"Start":"03:27.750 ","End":"03:30.554","Text":"so this would be our answer."},{"Start":"03:30.554 ","End":"03:35.645","Text":"Next one. First thing is to raise this product to the power of 3."},{"Start":"03:35.645 ","End":"03:37.730","Text":"Notice there are 3 things, there\u0027s the 2,"},{"Start":"03:37.730 ","End":"03:40.955","Text":"there\u0027s the c with its exponent and the d part."},{"Start":"03:40.955 ","End":"03:46.330","Text":"Each one of them has to be raised to the power of 3, so 2^3c."},{"Start":"03:46.330 ","End":"03:48.690","Text":"I\u0027m going to do 2 steps in 1,"},{"Start":"03:48.690 ","End":"03:50.850","Text":"c^-1 to the power of 3,"},{"Start":"03:50.850 ","End":"03:53.980","Text":"I\u0027m straightaway going to multiply minus 1 with 3,"},{"Start":"03:53.980 ","End":"03:56.705","Text":"because we\u0027re getting more of that so we can take more shortcuts."},{"Start":"03:56.705 ","End":"04:02.260","Text":"D to the power of minus 2 to the power of 3 is d to the minus 6."},{"Start":"04:02.260 ","End":"04:04.450","Text":"Here again, we have a product of 2 things,"},{"Start":"04:04.450 ","End":"04:07.280","Text":"each one of them has to be taken to the power of 4,"},{"Start":"04:07.280 ","End":"04:11.085","Text":"so 2^4, and then c^4."},{"Start":"04:11.085 ","End":"04:18.150","Text":"At this point, we just combine the 2\u0027s numbers then the c\u0027s and then the d\u0027s."},{"Start":"04:18.150 ","End":"04:22.085","Text":"As for 2\u0027s, we have 2^3, 2^4."},{"Start":"04:22.085 ","End":"04:25.835","Text":"Remember we add the exponents, 3+4 is 7."},{"Start":"04:25.835 ","End":"04:28.875","Text":"C, we have a minus 3 and a 4,"},{"Start":"04:28.875 ","End":"04:30.090","Text":"we add those,"},{"Start":"04:30.090 ","End":"04:32.355","Text":"so we get 4-3 is 1,"},{"Start":"04:32.355 ","End":"04:36.575","Text":"c^1 throughout the one if you like."},{"Start":"04:36.575 ","End":"04:40.810","Text":"Then d, just from here to the minus 6."},{"Start":"04:40.810 ","End":"04:46.519","Text":"Finally, I put everything with a negative exponent into the denominator."},{"Start":"04:46.519 ","End":"04:51.490","Text":"What I\u0027m left with is the d^6."},{"Start":"04:51.490 ","End":"04:56.270","Text":"Well, the minus 6 goes to become plus 6 in the denominator and the numerator,"},{"Start":"04:56.270 ","End":"04:59.885","Text":"the c, I just write it as c without the 1,"},{"Start":"04:59.885 ","End":"05:02.180","Text":"and numbers can be evaluated,"},{"Start":"05:02.180 ","End":"05:05.665","Text":"2^7 is 128,"},{"Start":"05:05.665 ","End":"05:09.050","Text":"and so this is the answer for this part."},{"Start":"05:09.050 ","End":"05:12.500","Text":"Then we have 2 more parts to do. Let\u0027s see."},{"Start":"05:12.500 ","End":"05:14.660","Text":"In this part,"},{"Start":"05:14.660 ","End":"05:16.115","Text":"I\u0027m going to first of all,"},{"Start":"05:16.115 ","End":"05:19.160","Text":"raise each fraction to its exponent."},{"Start":"05:19.160 ","End":"05:21.260","Text":"Remember when we raise a fraction to a power,"},{"Start":"05:21.260 ","End":"05:23.795","Text":"we do it for numerator and denominator."},{"Start":"05:23.795 ","End":"05:31.775","Text":"What we get is (m^-2)^3 which I write immediately as m^-6."},{"Start":"05:31.775 ","End":"05:34.670","Text":"Then here I have (n^3)^3,"},{"Start":"05:34.670 ","End":"05:40.000","Text":"which I write as n^9 because 3 times 3 is 9."},{"Start":"05:40.000 ","End":"05:44.700","Text":"(n^3)^2 is n^6,"},{"Start":"05:44.700 ","End":"05:47.325","Text":"because -3 times -2 is 6."},{"Start":"05:47.325 ","End":"05:50.640","Text":"Here, 2 times minus 2 is minus 4,"},{"Start":"05:50.640 ","End":"05:52.740","Text":"so m to the minus 4."},{"Start":"05:52.740 ","End":"05:55.665","Text":"Negative ones need to cross sides."},{"Start":"05:55.665 ","End":"06:01.865","Text":"In fact, perhaps it\u0027s best to just bring everything to the numerator I\u0027m now thinking,"},{"Start":"06:01.865 ","End":"06:06.080","Text":"so let\u0027s just put everything on the numerator and say m^-6,"},{"Start":"06:06.080 ","End":"06:09.080","Text":"n^-6, these 2 numerators."},{"Start":"06:09.080 ","End":"06:11.075","Text":"This one is a denominator,"},{"Start":"06:11.075 ","End":"06:14.720","Text":"so it\u0027s n^-9,"},{"Start":"06:14.720 ","End":"06:19.790","Text":"and this one when I cross sides becomes m^+4."},{"Start":"06:19.790 ","End":"06:24.265","Text":"I\u0027m going to combine m\u0027s together and then n\u0027s together."},{"Start":"06:24.265 ","End":"06:30.960","Text":"I\u0027ve got m^(-6+4) is -2 and then for the n,"},{"Start":"06:30.960 ","End":"06:35.670","Text":"I got 6-9 is minus 3."},{"Start":"06:35.670 ","End":"06:38.570","Text":"Both of these go into the denominator,"},{"Start":"06:38.570 ","End":"06:44.140","Text":"so the answer will be 1 over m squared n^3."},{"Start":"06:44.140 ","End":"06:46.854","Text":"Let\u0027s get the last one."},{"Start":"06:46.854 ","End":"06:50.600","Text":"In this one, I have a fraction to a power."},{"Start":"06:50.600 ","End":"06:53.690","Text":"I have to raise each one to that power."},{"Start":"06:53.690 ","End":"06:55.490","Text":"Let me just do that as the middle step."},{"Start":"06:55.490 ","End":"07:01.085","Text":"I\u0027ll just indicate that I want to take the numerator to the power of minus 2,"},{"Start":"07:01.085 ","End":"07:07.610","Text":"and then the denominator which is this also to the power of minus 2."},{"Start":"07:07.610 ","End":"07:09.350","Text":"When I have a product,"},{"Start":"07:09.350 ","End":"07:10.520","Text":"in this case, it\u0027s 3 things,"},{"Start":"07:10.520 ","End":"07:13.355","Text":"it\u0027s the 4 so the x squared and the y^-3,"},{"Start":"07:13.355 ","End":"07:16.070","Text":"I raise each of them to the power of minus 2."},{"Start":"07:16.070 ","End":"07:20.735","Text":"So 4^-2, x squared to the minus 2,"},{"Start":"07:20.735 ","End":"07:25.655","Text":"I use the rule that combines by multiplying the exponents."},{"Start":"07:25.655 ","End":"07:30.020","Text":"Then same here, minus 3 times minus 2 is 6,"},{"Start":"07:30.020 ","End":"07:38.835","Text":"so it\u0027s y^6 over (x^-4)^-2 is x^+8."},{"Start":"07:38.835 ","End":"07:44.065","Text":"Then because 5 - 2 is minus 10 we have here y^10."},{"Start":"07:44.065 ","End":"07:47.080","Text":"Now, I\u0027ll put everything on the numerator."},{"Start":"07:47.080 ","End":"07:48.760","Text":"That\u0027s one way to go about it."},{"Start":"07:48.760 ","End":"07:53.500","Text":"So 4^-2, x^-4."},{"Start":"07:53.500 ","End":"07:56.835","Text":"I\u0027ll just copy the numerator first of all, y^6."},{"Start":"07:56.835 ","End":"08:00.205","Text":"Then what\u0027s on the denominator reverses sign."},{"Start":"08:00.205 ","End":"08:05.735","Text":"X^-8, y^+10,"},{"Start":"08:05.735 ","End":"08:07.680","Text":"now I can combine."},{"Start":"08:07.680 ","End":"08:09.795","Text":"Numbers can be evaluated."},{"Start":"08:09.795 ","End":"08:15.535","Text":"4^-2 is 1/16."},{"Start":"08:15.535 ","End":"08:18.325","Text":"X^-4,"},{"Start":"08:18.325 ","End":"08:23.290","Text":"x^-8 is altogether x^-12."},{"Start":"08:23.290 ","End":"08:29.120","Text":"Here I have y^6, y^10 is y^16."},{"Start":"08:29.120 ","End":"08:32.185","Text":"What I\u0027m left with, let\u0027s see,"},{"Start":"08:32.185 ","End":"08:36.345","Text":"in the numerator I have y^16,"},{"Start":"08:36.345 ","End":"08:43.420","Text":"and then the denominator we have this 16 and x^-12 is 1 over x^12."},{"Start":"08:43.420 ","End":"08:48.415","Text":"Usually, always should write your answer as 1 single fraction,"},{"Start":"08:48.415 ","End":"08:52.735","Text":"not like a 16th times 1 over x^12, combine everything."},{"Start":"08:52.735 ","End":"08:56.905","Text":"That\u0027s 1 single fraction and no negative exponents here, so we\u0027re fine."},{"Start":"08:56.905 ","End":"09:00.950","Text":"That\u0027s it for this whole exercise."}],"ID":8106},{"Watched":false,"Name":"Exercise 3","Duration":"10m 47s","ChapterTopicVideoID":8014,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8014.jpeg","UploadDate":"2020-09-30T14:19:43.8530000","DurationForVideoObject":"PT10M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"This exercise is made up of 4 separate exercises"},{"Start":"00:03.960 ","End":"00:07.980","Text":"and they\u0027ve thrown the works and we have fractional exponents,"},{"Start":"00:07.980 ","End":"00:09.390","Text":"we have letters,"},{"Start":"00:09.390 ","End":"00:13.185","Text":"we have fractions, negatives, everything."},{"Start":"00:13.185 ","End":"00:16.050","Text":"They look scary but let\u0027s just take them one at"},{"Start":"00:16.050 ","End":"00:19.575","Text":"a time and we\u0027ll apply the rules carefully"},{"Start":"00:19.575 ","End":"00:22.865","Text":"and you\u0027ll see that they end up simply and"},{"Start":"00:22.865 ","End":"00:26.790","Text":"it\u0027s not that difficult if you just take it nice and easy."},{"Start":"00:26.790 ","End":"00:30.075","Text":"Let\u0027s start with part A."},{"Start":"00:30.075 ","End":"00:33.020","Text":"The first thing I see is something to the power of"},{"Start":"00:33.020 ","End":"00:36.695","Text":"1/2 or something to an exponent and it\u0027s a fraction."},{"Start":"00:36.695 ","End":"00:40.535","Text":"I\u0027m going to use the rule where when you have a fraction to a power,"},{"Start":"00:40.535 ","End":"00:43.790","Text":"you do it to numerator and denominator separately."},{"Start":"00:43.790 ","End":"00:47.800","Text":"So I\u0027ve got (a^ minus 2)^1/2,"},{"Start":"00:47.800 ","End":"00:49.640","Text":"and then a dividing line."},{"Start":"00:49.640 ","End":"00:54.260","Text":"Then notice that the denominator is a product of 2 separate things,"},{"Start":"00:54.260 ","End":"00:56.654","Text":"the 4 and the v to the power of."},{"Start":"00:56.654 ","End":"00:59.315","Text":"So each of them has to be raised to the power of a 1/2."},{"Start":"00:59.315 ","End":"01:01.130","Text":"I have 4^1/2."},{"Start":"01:01.130 ","End":"01:04.400","Text":"I emphasize this by the way because that\u0027s a common mistake is to forget"},{"Start":"01:04.400 ","End":"01:08.000","Text":"about raising the number also to the power. So I\u0027m pointing that out."},{"Start":"01:08.000 ","End":"01:11.480","Text":"Don\u0027t make that mistake if you do remember to do that."},{"Start":"01:11.480 ","End":"01:16.850","Text":"Then (v^ minus 1/2)^1/2."},{"Start":"01:16.850 ","End":"01:20.945","Text":"Now, we can simplify each of the bits like here,"},{"Start":"01:20.945 ","End":"01:22.255","Text":"a power of a power,"},{"Start":"01:22.255 ","End":"01:23.689","Text":"so we multiply the powers."},{"Start":"01:23.689 ","End":"01:27.035","Text":"Minus 2 times a 1/2 is minus 1."},{"Start":"01:27.035 ","End":"01:29.615","Text":"Here we have a^ minus 1."},{"Start":"01:29.615 ","End":"01:33.605","Text":"That\u0027s the numerator. Denominator, 4 is a number."},{"Start":"01:33.605 ","End":"01:35.615","Text":"4^1/2 is a computation."},{"Start":"01:35.615 ","End":"01:39.055","Text":"It means the square root of 4, which is 2."},{"Start":"01:39.055 ","End":"01:41.300","Text":"It means the positive square root."},{"Start":"01:41.300 ","End":"01:43.220","Text":"The positive number whose square is 4, that\u0027s 2."},{"Start":"01:43.220 ","End":"01:47.435","Text":"Here once again, we have a product of exponents that we have to do."},{"Start":"01:47.435 ","End":"01:53.285","Text":"It\u0027s v^ minus 1/2 times a 1/2 is minus 1/4."},{"Start":"01:53.285 ","End":"01:55.580","Text":"Now, how do we proceed from here?"},{"Start":"01:55.580 ","End":"01:59.300","Text":"I don\u0027t want negatives so each time there\u0027s a negative,"},{"Start":"01:59.300 ","End":"02:02.795","Text":"it will cross the boundary and become a positive."},{"Start":"02:02.795 ","End":"02:04.940","Text":"Here\u0027s my dividing line."},{"Start":"02:04.940 ","End":"02:07.205","Text":"The 2 stays down here."},{"Start":"02:07.205 ","End":"02:11.645","Text":"The a comes down and becomes a^1."},{"Start":"02:11.645 ","End":"02:18.065","Text":"The v bit goes to the top and it becomes v^1/4."},{"Start":"02:18.065 ","End":"02:23.510","Text":"When we simplify, we generally prefer a root rather than a fractional exponent,"},{"Start":"02:23.510 ","End":"02:26.900","Text":"so I\u0027ll write this as the fourth root of"},{"Start":"02:26.900 ","End":"02:31.550","Text":"v and the denominator is the same except that I check out the 1,"},{"Start":"02:31.550 ","End":"02:33.400","Text":"and we don\u0027t put to the power of 1."},{"Start":"02:33.400 ","End":"02:36.140","Text":"That\u0027s the first one. Not too bad."},{"Start":"02:36.140 ","End":"02:38.435","Text":"Let\u0027s try the second. Let\u0027s see."},{"Start":"02:38.435 ","End":"02:41.869","Text":"Once again, we have a fraction raised to a power."},{"Start":"02:41.869 ","End":"02:46.975","Text":"I\u0027m going to take a bit of shortcuts here because we\u0027re already sophisticated."},{"Start":"02:46.975 ","End":"02:49.910","Text":"I\u0027m going to take the numerator to the power of 1/5,"},{"Start":"02:49.910 ","End":"02:51.350","Text":"but the numerator is a product,"},{"Start":"02:51.350 ","End":"02:54.020","Text":"so I\u0027m going to take each bit to the power of 1/5."},{"Start":"02:54.020 ","End":"02:57.605","Text":"I have (x^1/2)^1/5."},{"Start":"02:57.605 ","End":"03:01.070","Text":"I almost did the product also in one step."},{"Start":"03:01.070 ","End":"03:03.915","Text":"This is already 2 steps in 1,"},{"Start":"03:03.915 ","End":"03:06.510","Text":"then the second part to the power of a 1/5,"},{"Start":"03:06.510 ","End":"03:13.850","Text":"which is (y^ minus 5)^1/5.Then the denominator also to the power of a 1/5,"},{"Start":"03:13.850 ","End":"03:15.860","Text":"but the denominator is a product."},{"Start":"03:15.860 ","End":"03:23.160","Text":"So I take (y^ minus 25)^1/5,"},{"Start":"03:23.160 ","End":"03:27.120","Text":"and then (x^ minus 3)^1/5."},{"Start":"03:27.120 ","End":"03:30.260","Text":"There\u0027s a rule of thumb that if I have not just quotients and products,"},{"Start":"03:30.260 ","End":"03:33.575","Text":"if I have a mixture of products and quotients like this,"},{"Start":"03:33.575 ","End":"03:35.720","Text":"what we\u0027ll have to do is take each of the pieces."},{"Start":"03:35.720 ","End":"03:37.010","Text":"In this case I have 4 pieces,"},{"Start":"03:37.010 ","End":"03:39.011","Text":"this one, this one, this one, and this one,"},{"Start":"03:39.011 ","End":"03:41.420","Text":"and each of the pieces I raise to the power of 1/5,"},{"Start":"03:41.420 ","End":"03:43.940","Text":"provided it\u0027s all multiplications and divisions."},{"Start":"03:43.940 ","End":"03:46.205","Text":"That\u0027s another rule of thumb."},{"Start":"03:46.205 ","End":"03:50.810","Text":"Now, each of these 4 bits is power of a power."},{"Start":"03:50.810 ","End":"03:53.360","Text":"So we multiply the exponents, in this case,"},{"Start":"03:53.360 ","End":"03:57.275","Text":"minus a 1/2 times a 1/5 is minus a 1/10."},{"Start":"03:57.275 ","End":"04:02.740","Text":"Then minus 5 times 1/5 is minus 1."},{"Start":"04:02.740 ","End":"04:07.385","Text":"Here we have to do 25 over 5 rather with a minus."},{"Start":"04:07.385 ","End":"04:09.590","Text":"Anyway, it\u0027s minus 5."},{"Start":"04:09.590 ","End":"04:13.895","Text":"Here x to the minus 3/5."},{"Start":"04:13.895 ","End":"04:17.180","Text":"Now as usual, there\u0027s several ways of proceeding."},{"Start":"04:17.180 ","End":"04:19.880","Text":"I\u0027ll just say, let\u0027s put everything in the numerator for"},{"Start":"04:19.880 ","End":"04:22.820","Text":"a moment because I want to combine the x\u0027s,"},{"Start":"04:22.820 ","End":"04:25.240","Text":"I want to combine the y\u0027s."},{"Start":"04:25.240 ","End":"04:29.900","Text":"Let me say that this is x to the minus a 1/10,"},{"Start":"04:29.900 ","End":"04:37.704","Text":"y to the minus 1. y^5 because when we cross the dividing line,"},{"Start":"04:37.704 ","End":"04:43.880","Text":"then we reverse the sign of the exponent and this becomes x^3/5."},{"Start":"04:43.880 ","End":"04:50.630","Text":"Now, I want to take the x\u0027s and combine them using the rule that this times this,"},{"Start":"04:50.630 ","End":"04:54.935","Text":"multiplication means addition as far as the exponents go."},{"Start":"04:54.935 ","End":"04:57.665","Text":"Let me do this fraction at the side,"},{"Start":"04:57.665 ","End":"04:59.735","Text":"3/5 minus a 1/10."},{"Start":"04:59.735 ","End":"05:03.185","Text":"Let\u0027s see if I put it all over 10, I\u0027ve got 6."},{"Start":"05:03.185 ","End":"05:08.910","Text":"That\u0027s because this is 6/10 minus 1/10 is 5/10, it\u0027s a 1/2."},{"Start":"05:10.180 ","End":"05:13.460","Text":"x^1.5. Here we have whole numbers for y."},{"Start":"05:13.460 ","End":"05:16.970","Text":"5 minus 1 is 4, y^4."},{"Start":"05:16.970 ","End":"05:20.825","Text":"Finally, I\u0027ll convert the fractional exponent to a root."},{"Start":"05:20.825 ","End":"05:25.220","Text":"The second root is the square root, is just root."},{"Start":"05:25.220 ","End":"05:29.255","Text":"Square root of x times y^4."},{"Start":"05:29.255 ","End":"05:32.945","Text":"The square root ends here, y is separate."},{"Start":"05:32.945 ","End":"05:35.615","Text":"That\u0027s the answer to b. We are getting there."},{"Start":"05:35.615 ","End":"05:40.400","Text":"Next, go on to part c and let\u0027s see about"},{"Start":"05:40.400 ","End":"05:43.610","Text":"c. I\u0027m going to use that rule of thumb I told you earlier"},{"Start":"05:43.610 ","End":"05:46.820","Text":"that when I have multiplications and divisions,"},{"Start":"05:46.820 ","End":"05:50.450","Text":"I can just take each bit and raise it to the power of a 1/6."},{"Start":"05:50.450 ","End":"05:59.470","Text":"I have (a^ minus)^1^6 times (b^1/2)^1/6."},{"Start":"05:59.470 ","End":"06:09.640","Text":"Dividing line, (c^3)^1/6, and (d^minus2)^1/6."},{"Start":"06:09.740 ","End":"06:15.380","Text":"Let\u0027s see, we don\u0027t have any letters in common in top and bottom."},{"Start":"06:15.380 ","End":"06:18.680","Text":"Just noting that, have an impact on my strategy."},{"Start":"06:18.680 ","End":"06:23.300","Text":"First of all, I\u0027ll just go to multiply the exponents because we have a power of a power."},{"Start":"06:23.300 ","End":"06:26.390","Text":"Let\u0027s do this and get the 4 bits expanded."},{"Start":"06:26.390 ","End":"06:30.795","Text":"(A^ minus 2)^1/6 is minus a 1/3,"},{"Start":"06:30.795 ","End":"06:33.230","Text":"I\u0027m just going to write to multiply fractions."},{"Start":"06:33.230 ","End":"06:35.716","Text":"b to the power of 1/12,"},{"Start":"06:35.716 ","End":"06:37.610","Text":"this is not clear."},{"Start":"06:37.610 ","End":"06:41.450","Text":"That\u0027s better, over, c to the,"},{"Start":"06:41.450 ","End":"06:46.940","Text":"it\u0027s obviously a 1/2 and this is obviously minus a 1/3."},{"Start":"06:46.940 ","End":"06:48.800","Text":"There\u0027s nothing to combine here."},{"Start":"06:48.800 ","End":"06:51.050","Text":"What I\u0027m going to do is just throw the ones with"},{"Start":"06:51.050 ","End":"06:54.140","Text":"the negative exponents to the other side."},{"Start":"06:54.140 ","End":"06:59.035","Text":"This one\u0027s going to cross over and this one with a minus is going to cross over."},{"Start":"06:59.035 ","End":"07:00.870","Text":"What I\u0027m going to be left with,"},{"Start":"07:00.870 ","End":"07:04.000","Text":"let\u0027s see, b^1/12."},{"Start":"07:04.100 ","End":"07:09.725","Text":"Then I\u0027ve got the d coming up, so that\u0027s d^1/3."},{"Start":"07:09.725 ","End":"07:14.540","Text":"On the bottom, I have a^1/3 without"},{"Start":"07:14.540 ","End":"07:20.345","Text":"the minus and the c was already here to the power of a 1/2."},{"Start":"07:20.345 ","End":"07:22.610","Text":"Now I\u0027ll just write it,"},{"Start":"07:22.610 ","End":"07:26.690","Text":"instead of with fractional exponents as roots."},{"Start":"07:26.690 ","End":"07:31.250","Text":"12th root of b^3 root,"},{"Start":"07:31.250 ","End":"07:35.970","Text":"or third root of d divided"},{"Start":"07:35.970 ","End":"07:41.960","Text":"by cube root of a and square root."},{"Start":"07:41.960 ","End":"07:43.850","Text":"You never say second root,"},{"Start":"07:43.850 ","End":"07:47.540","Text":"square root of c and we don\u0027t need to write the 2 there."},{"Start":"07:47.540 ","End":"07:52.685","Text":"That\u0027s the answer to part c and we\u0027re on the last one now."},{"Start":"07:52.685 ","End":"07:57.590","Text":"As before, we\u0027re going to take each bit to the power of a 1/3."},{"Start":"07:57.590 ","End":"07:59.320","Text":"But notice that here I have 3 bits."},{"Start":"07:59.320 ","End":"08:02.735","Text":"The 2 is also one of the factors."},{"Start":"08:02.735 ","End":"08:07.495","Text":"2^1/3, (c^ minus 1)^1/3, 1/3 everywhere."},{"Start":"08:07.495 ","End":"08:10.305","Text":"This to the power of a 1/3."},{"Start":"08:10.305 ","End":"08:14.625","Text":"The same here,"},{"Start":"08:14.625 ","End":"08:17.085","Text":"2^ this time it\u0027s 1/4, and this one,"},{"Start":"08:17.085 ","End":"08:20.550","Text":"c to the minus 4 also to the power of a 1/4."},{"Start":"08:20.550 ","End":"08:23.190","Text":"Let\u0027s see what that gives me."},{"Start":"08:23.190 ","End":"08:28.621","Text":"2^1/3, I\u0027ll just leave it like that meanwhile."},{"Start":"08:28.621 ","End":"08:35.990","Text":"c^1/3, d^ minus 2/3."},{"Start":"08:35.990 ","End":"08:38.225","Text":"What do I have in the denominator?"},{"Start":"08:38.225 ","End":"08:41.570","Text":"I\u0027ll just leave that as 2^1/4,"},{"Start":"08:41.570 ","End":"08:44.120","Text":"power of a fourth and then c to the power of,"},{"Start":"08:44.120 ","End":"08:47.735","Text":"let\u0027s see, minus 4 times a 1/4 is minus 1."},{"Start":"08:47.735 ","End":"08:50.690","Text":"Now, I do want to combine things."},{"Start":"08:50.690 ","End":"08:55.415","Text":"I have the 2 in common and the c in common."},{"Start":"08:55.415 ","End":"09:04.445","Text":"I\u0027ll write this as 2^1/3 minus a 1/4."},{"Start":"09:04.445 ","End":"09:06.499","Text":"Let\u0027s just say we\u0027ve done this division."},{"Start":"09:06.499 ","End":"09:10.285","Text":"Then c^1/3."},{"Start":"09:10.285 ","End":"09:13.220","Text":"It\u0027s going to be plus 1 because if I brought this over to the top,"},{"Start":"09:13.220 ","End":"09:14.585","Text":"it would be plus 1."},{"Start":"09:14.585 ","End":"09:16.685","Text":"Or if you like, it\u0027s minus a 1/3,"},{"Start":"09:16.685 ","End":"09:18.050","Text":"less minus 1,"},{"Start":"09:18.050 ","End":"09:19.430","Text":"which means plus 1."},{"Start":"09:19.430 ","End":"09:22.175","Text":"The t is just what it is."},{"Start":"09:22.175 ","End":"09:24.919","Text":"What can we simplify further?"},{"Start":"09:24.919 ","End":"09:27.350","Text":"Certainly, we can compute the fraction."},{"Start":"09:27.350 ","End":"09:29.555","Text":"A 1/3 minus a 1/4 is a 1/12."},{"Start":"09:29.555 ","End":"09:32.330","Text":"I\u0027ll leave it to you to figure out."},{"Start":"09:32.330 ","End":"09:35.345","Text":"I think of it in terms of the very large pizza,"},{"Start":"09:35.345 ","End":"09:39.350","Text":"12 slices a 1/3 of it is 4 slices,"},{"Start":"09:39.350 ","End":"09:43.310","Text":"a 1/4 of it is 3 slices like a 1 slice pizza model."},{"Start":"09:43.310 ","End":"09:49.040","Text":"Here we have 1 minus a 1/3 is 2/3 and here we have a minus."},{"Start":"09:49.040 ","End":"09:52.145","Text":"That we want to put into the denominator."},{"Start":"09:52.145 ","End":"09:54.860","Text":"That\u0027s d to the power of 2/3."},{"Start":"09:54.860 ","End":"09:58.310","Text":"But I want to go one step further because we"},{"Start":"09:58.310 ","End":"10:03.530","Text":"prefer roots to fractional exponents considered more simplified."},{"Start":"10:03.530 ","End":"10:05.540","Text":"Anyway, just how it is."},{"Start":"10:05.540 ","End":"10:11.640","Text":"So 12th root of 2 and then c^2/3,"},{"Start":"10:11.640 ","End":"10:16.560","Text":"either the cube root of c^2 or cube root of c, and then squared."},{"Start":"10:16.560 ","End":"10:19.975","Text":"I\u0027ll write it as the cube root of c^2."},{"Start":"10:19.975 ","End":"10:22.850","Text":"When I\u0027m doing computations as we did with numbers,"},{"Start":"10:22.850 ","End":"10:25.445","Text":"I prefer to first take the cube root and then square it."},{"Start":"10:25.445 ","End":"10:27.020","Text":"When you\u0027re dealing with variables,"},{"Start":"10:27.020 ","End":"10:30.290","Text":"it looks nicer if you first square it and then take the cube root."},{"Start":"10:30.290 ","End":"10:32.180","Text":"That\u0027s aesthetics really,"},{"Start":"10:32.180 ","End":"10:34.445","Text":"what is called simplifying."},{"Start":"10:34.445 ","End":"10:38.160","Text":"Here also power of 2/3."},{"Start":"10:38.160 ","End":"10:43.360","Text":"I\u0027ll take first of all to the power of 2 and then cube root it."},{"Start":"10:43.360 ","End":"10:47.820","Text":"That\u0027s the answer to the last part and we are done."}],"ID":8107},{"Watched":false,"Name":"Exercise 4","Duration":"6m 56s","ChapterTopicVideoID":8015,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8015.jpeg","UploadDate":"2020-09-30T14:24:54.1100000","DurationForVideoObject":"PT6M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"This exercise is 81."},{"Start":"00:03.150 ","End":"00:07.605","Text":"The common theme is that they all involve roots are radicals,"},{"Start":"00:07.605 ","End":"00:09.930","Text":"and all of the numerical."},{"Start":"00:09.930 ","End":"00:15.270","Text":"And actually, all can be done without a calculator or I should have even added,"},{"Start":"00:15.270 ","End":"00:18.480","Text":"without a calculator if possible."},{"Start":"00:18.480 ","End":"00:21.780","Text":"We get to part a,"},{"Start":"00:21.780 ","End":"00:24.580","Text":"fourth root of 16."},{"Start":"00:25.010 ","End":"00:29.960","Text":"Fourth root of 16 you should remember it means what to the power of 4 gives 16."},{"Start":"00:29.960 ","End":"00:32.660","Text":"We\u0027ve seen this often enough and it\u0027s 2."},{"Start":"00:32.660 ","End":"00:37.310","Text":"But I\u0027ll write that explanation at the side because 2^4 is 16,"},{"Start":"00:37.310 ","End":"00:39.745","Text":"that we say the fourth root of 16 is 2."},{"Start":"00:39.745 ","End":"00:43.770","Text":"Now it\u0027s also true that minus 2^4 is16."},{"Start":"00:43.770 ","End":"00:46.195","Text":"But when we just write a root,"},{"Start":"00:46.195 ","End":"00:50.330","Text":"we mean the positive root where possible."},{"Start":"00:50.330 ","End":"00:51.920","Text":"Now with a cube root,"},{"Start":"00:51.920 ","End":"00:53.270","Text":"there is no doubt."},{"Start":"00:53.270 ","End":"00:58.840","Text":"What I want to know is what number it is here that\u0027s the power of 3."},{"Start":"00:58.840 ","End":"01:02.865","Text":"Something to the power of 3 equals 125."},{"Start":"01:02.865 ","End":"01:07.040","Text":"Again it\u0027s 1 of those things that keeps turning up and it\u0027s 5^3."},{"Start":"01:07.040 ","End":"01:12.495","Text":"And that we can do in our heads 5 times 5 is 25 times 5 is 125."},{"Start":"01:12.495 ","End":"01:15.985","Text":"So the answer to this is 5."},{"Start":"01:15.985 ","End":"01:20.165","Text":"What the remark I made it in part a is not even applicable here"},{"Start":"01:20.165 ","End":"01:25.055","Text":"because (minus 5)^3 is not a 125."},{"Start":"01:25.055 ","End":"01:28.755","Text":"With the even numbers is always this confusion,"},{"Start":"01:28.755 ","End":"01:31.130","Text":"some people get confused between"},{"Start":"01:31.130 ","End":"01:35.525","Text":"the solution to what to the power of 4 is 16 and the fourth root."},{"Start":"01:35.525 ","End":"01:39.685","Text":"Fourth root also implies that you take the positive where possible."},{"Start":"01:39.685 ","End":"01:43.250","Text":"I hope I clarified something rather than confused."},{"Start":"01:43.250 ","End":"01:45.170","Text":"Anyway, let\u0027s go on."},{"Start":"01:45.170 ","End":"01:47.540","Text":"Here we have a number here and a number here."},{"Start":"01:47.540 ","End":"01:50.765","Text":"And according to the definition of fractional exponents,"},{"Start":"01:50.765 ","End":"01:55.700","Text":"now, when I have an exponent under the radical,"},{"Start":"01:55.700 ","End":"02:00.560","Text":"the way it\u0027s expressed as a fractional exponent is by saying"},{"Start":"02:00.560 ","End":"02:05.420","Text":"64 to the power of the number here over the number here."},{"Start":"02:05.420 ","End":"02:10.280","Text":"The reason I\u0027m taking it to a fractional exponents is because,"},{"Start":"02:10.280 ","End":"02:16.175","Text":"this has another interpretation where instead of squaring and then taking the cube root,"},{"Start":"02:16.175 ","End":"02:18.470","Text":"I can do it in the opposite order."},{"Start":"02:18.470 ","End":"02:20.090","Text":"I can take first of all,"},{"Start":"02:20.090 ","End":"02:22.385","Text":"the cube root of 64."},{"Start":"02:22.385 ","End":"02:24.185","Text":"And then the answer to that,"},{"Start":"02:24.185 ","End":"02:26.525","Text":"I can raise to the power of 2."},{"Start":"02:26.525 ","End":"02:28.655","Text":"It didn\u0027t actually need this middle step."},{"Start":"02:28.655 ","End":"02:31.700","Text":"You can remember that when you have a root and a power,"},{"Start":"02:31.700 ","End":"02:33.005","Text":"you can do them in either order."},{"Start":"02:33.005 ","End":"02:34.955","Text":"Now why do I want to change the order?"},{"Start":"02:34.955 ","End":"02:37.775","Text":"Just practically because I don\u0027t have a calculator."},{"Start":"02:37.775 ","End":"02:39.710","Text":"And if you take the cube root first,"},{"Start":"02:39.710 ","End":"02:41.120","Text":"you get smaller numbers."},{"Start":"02:41.120 ","End":"02:45.935","Text":"64 squared is maybe not that easy to compute without a calculator."},{"Start":"02:45.935 ","End":"02:49.219","Text":"So this way round the cube root of 64,"},{"Start":"02:49.219 ","End":"02:53.200","Text":"we\u0027ve seen so many times that this is equal to 4."},{"Start":"02:53.200 ","End":"03:01.035","Text":"That\u0027s just the cube root is 4 cubed is 64 because 4 times 4 times 4 is 64."},{"Start":"03:01.035 ","End":"03:02.750","Text":"And then I have the 2 here."},{"Start":"03:02.750 ","End":"03:06.380","Text":"And so the answer is 16."},{"Start":"03:06.380 ","End":"03:09.000","Text":"But if you did 64 squared,"},{"Start":"03:09.000 ","End":"03:16.885","Text":"I think it comes out to be 4,096 and then go figure the cube root of that."},{"Start":"03:16.885 ","End":"03:18.990","Text":"16 times 16 times 16,"},{"Start":"03:18.990 ","End":"03:20.810","Text":"but it\u0027s not one of the immediate ones."},{"Start":"03:20.810 ","End":"03:22.805","Text":"So if you don\u0027t have a calculator,"},{"Start":"03:22.805 ","End":"03:24.605","Text":"better to take the root first."},{"Start":"03:24.605 ","End":"03:26.360","Text":"The same thing in d, I\u0027m going to use"},{"Start":"03:26.360 ","End":"03:29.435","Text":"that same lesson as we did in policy on it this time,"},{"Start":"03:29.435 ","End":"03:32.225","Text":"I\u0027m going to write it straight away as first,"},{"Start":"03:32.225 ","End":"03:38.510","Text":"I wanted to do the fifth root of 32 and the answer to that raised to the power of 4."},{"Start":"03:38.510 ","End":"03:42.125","Text":"Now fifth root of 32, we\u0027ve seen this a dozen times,"},{"Start":"03:42.125 ","End":"03:47.045","Text":"it\u0027s 2 because 2^5 is 32, 2^1,"},{"Start":"03:47.045 ","End":"03:48.770","Text":"keep doubling up, 2^2,"},{"Start":"03:48.770 ","End":"03:52.955","Text":"2^3, 2^4, 2^5."},{"Start":"03:52.955 ","End":"03:57.720","Text":"So in case you\u0027re counting on your fingers and power of 1,2,3,4,5."},{"Start":"03:57.720 ","End":"04:00.555","Text":"So that\u0027s fine this thing is 2."},{"Start":"04:00.555 ","End":"04:04.685","Text":"Then we have the 4 here and comes out to be 16,"},{"Start":"04:04.685 ","End":"04:07.010","Text":"same as the previous 1, coincidence."},{"Start":"04:07.010 ","End":"04:09.640","Text":"Cube root of minus 8."},{"Start":"04:09.640 ","End":"04:11.150","Text":"Now, in this case,"},{"Start":"04:11.150 ","End":"04:15.110","Text":"I want to know what number to the power of 3 is minus 8."},{"Start":"04:15.110 ","End":"04:17.135","Text":"Notice that 3 is an odd number."},{"Start":"04:17.135 ","End":"04:20.960","Text":"I\u0027m going to taking an odd root of a negative number, That\u0027s fine."},{"Start":"04:20.960 ","End":"04:24.350","Text":"If we had an even order root of a negative number,"},{"Start":"04:24.350 ","End":"04:25.730","Text":"then it\u0027s a problem."},{"Start":"04:25.730 ","End":"04:31.150","Text":"So here we\u0027re basically asking what number cubed is minus 8."},{"Start":"04:31.150 ","End":"04:36.825","Text":"First of all, ask what number to the power of 3 is 8?"},{"Start":"04:36.825 ","End":"04:39.775","Text":"The answer to that is 2,"},{"Start":"04:39.775 ","End":"04:41.255","Text":"once I have that,"},{"Start":"04:41.255 ","End":"04:44.210","Text":"then I know that my answer is not 2 but minus"},{"Start":"04:44.210 ","End":"04:47.435","Text":"2 because an odd number of minuses is minus."},{"Start":"04:47.435 ","End":"04:50.270","Text":"So this is minus 2, like I said,"},{"Start":"04:50.270 ","End":"04:53.990","Text":"the root means if it\u0027s positive 1,"},{"Start":"04:53.990 ","End":"04:56.059","Text":"if it\u0027s ambiguous, if it\u0027s not ambiguous,"},{"Start":"04:56.059 ","End":"04:58.550","Text":"then the root is negative."},{"Start":"04:58.550 ","End":"05:03.990","Text":"Minus 2 times minus 2 times minus 2 is minus 8 because like I said,"},{"Start":"05:03.990 ","End":"05:05.450","Text":"2 times 2 times 2 is 8 minus,"},{"Start":"05:05.450 ","End":"05:07.055","Text":"minus, minus is minus."},{"Start":"05:07.055 ","End":"05:08.810","Text":"Let\u0027s see, where are we now,"},{"Start":"05:08.810 ","End":"05:10.820","Text":"we have still f,"},{"Start":"05:10.820 ","End":"05:12.485","Text":"g, and h to do."},{"Start":"05:12.485 ","End":"05:15.110","Text":"This is a similar 1 to the previous."},{"Start":"05:15.110 ","End":"05:17.555","Text":"We have once again,"},{"Start":"05:17.555 ","End":"05:21.574","Text":"an odd order root of a negative number."},{"Start":"05:21.574 ","End":"05:28.730","Text":"So if I found out what the fifth root of 32 was and ask what number to the fifth is 32?"},{"Start":"05:28.730 ","End":"05:32.030","Text":"Well then now it\u0027s too, we\u0027ve seen this even here."},{"Start":"05:32.030 ","End":"05:34.325","Text":"2^5 is 32."},{"Start":"05:34.325 ","End":"05:39.875","Text":"So I know that minus 2 to the fifth will be minus 32."},{"Start":"05:39.875 ","End":"05:44.030","Text":"And I\u0027m relying on the fact that 5 is an odd number because minus, minus, minus,"},{"Start":"05:44.030 ","End":"05:48.770","Text":"minus, minus the oddness makes the odd number of minuses gives us the minus."},{"Start":"05:48.770 ","End":"05:51.155","Text":"Here we have minus 2."},{"Start":"05:51.155 ","End":"05:53.450","Text":"The same idea again here."},{"Start":"05:53.450 ","End":"05:59.375","Text":"If I do the cube root to 27 because 3^3 is 27."},{"Start":"05:59.375 ","End":"06:03.290","Text":"I know that if I put a minus 3 to the power of 3, and again,"},{"Start":"06:03.290 ","End":"06:04.670","Text":"3 is an odd number,"},{"Start":"06:04.670 ","End":"06:07.790","Text":"this is important, I get minus 27."},{"Start":"06:07.790 ","End":"06:12.020","Text":"So the answer here is just minus 3 because it answers the question,"},{"Start":"06:12.020 ","End":"06:14.675","Text":"what to the power of 3 is minus 27."},{"Start":"06:14.675 ","End":"06:17.810","Text":"And here also odd number and"},{"Start":"06:17.810 ","End":"06:21.935","Text":"a negative business about it being an odd number is very important."},{"Start":"06:21.935 ","End":"06:24.995","Text":"If I had a sixth root or a fourth root or something,"},{"Start":"06:24.995 ","End":"06:26.210","Text":"there would be no answer."},{"Start":"06:26.210 ","End":"06:30.395","Text":"There\u0027s no even order route for a negative number with odds were okay."},{"Start":"06:30.395 ","End":"06:36.325","Text":"Basically we just, first of all ignore the minus fifth root of 243 is 3."},{"Start":"06:36.325 ","End":"06:40.100","Text":"That\u0027s because 3 to the 5 is 243."},{"Start":"06:40.100 ","End":"06:41.420","Text":"I know we\u0027ve seen this before."},{"Start":"06:41.420 ","End":"06:44.480","Text":"And so the answer to this will be minus 3."},{"Start":"06:44.480 ","End":"06:47.585","Text":"If I put a minus 3^5,"},{"Start":"06:47.585 ","End":"06:53.290","Text":"that will give me minus to the fifth is minus 243."},{"Start":"06:53.290 ","End":"06:56.440","Text":"That\u0027s the last part we\u0027re done."}],"ID":8108},{"Watched":false,"Name":"Exercise 5","Duration":"9m 30s","ChapterTopicVideoID":8016,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8016.jpeg","UploadDate":"2020-09-30T14:30:16.2030000","DurationForVideoObject":"PT9M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"This exercise is made up of 9 separate parts."},{"Start":"00:04.740 ","End":"00:08.550","Text":"This is practice for exponents which"},{"Start":"00:08.550 ","End":"00:12.690","Text":"are fractional and these are all numerical calculations,"},{"Start":"00:12.690 ","End":"00:14.505","Text":"there\u0027s no variables in this one."},{"Start":"00:14.505 ","End":"00:21.263","Text":"Anyway, let\u0027s get started with part A which is 9^1/2,"},{"Start":"00:21.263 ","End":"00:26.700","Text":"and something to the power of a half is the square root of 9."},{"Start":"00:26.700 ","End":"00:29.655","Text":"Normally, when we have to the power of 1,"},{"Start":"00:29.655 ","End":"00:31.095","Text":"we write the nth square root of,"},{"Start":"00:31.095 ","End":"00:33.405","Text":"but we don\u0027t write the 2 here."},{"Start":"00:33.405 ","End":"00:35.415","Text":"We could write a 2 here,"},{"Start":"00:35.415 ","End":"00:37.380","Text":"but it\u0027s not customary."},{"Start":"00:37.380 ","End":"00:43.500","Text":"The square root of 9 means what positive number squared gives me 9?"},{"Start":"00:43.500 ","End":"00:46.451","Text":"The answer is 3 and only 3,"},{"Start":"00:46.451 ","End":"00:48.960","Text":"minus 3 is not the square root of 9."},{"Start":"00:48.960 ","End":"00:53.780","Text":"The square root means the positive number such that the square is 9."},{"Start":"00:53.780 ","End":"00:57.825","Text":"Next one. This time we have a power of a 1/4,"},{"Start":"00:57.825 ","End":"00:59.880","Text":"that means the 4th root, in this case,"},{"Start":"00:59.880 ","End":"01:01.190","Text":"we have to write the 4,"},{"Start":"01:01.190 ","End":"01:09.650","Text":"4th root of 16 and this means the positive number such that to the power 4 is 16"},{"Start":"01:09.650 ","End":"01:14.330","Text":"so it\u0027s just one of those well known ones and the answer is"},{"Start":"01:14.330 ","End":"01:20.765","Text":"2 and maybe I\u0027ll write the explanation because 2^4 is 16 and check it."},{"Start":"01:20.765 ","End":"01:24.000","Text":"You don\u0027t need a calculator 2 times 2 times 2 times 2 is 16."},{"Start":"01:24.000 ","End":"01:28.740","Text":"Just like here, the reasoning for this was that 3^2 is equals to 9 and that"},{"Start":"01:28.740 ","End":"01:34.140","Text":"justifies it and we want the positive 1 because minus 3^2 also is 9,"},{"Start":"01:34.140 ","End":"01:37.890","Text":"minus 2^4 also is 16 but when we write this,"},{"Start":"01:37.890 ","End":"01:39.030","Text":"we mean the positive."},{"Start":"01:39.030 ","End":"01:44.251","Text":"cube root, power of a 1/3 is cube root and that we write with a 3 here."},{"Start":"01:44.251 ","End":"01:46.995","Text":"cube root of 81,"},{"Start":"01:46.995 ","End":"01:49.555","Text":"but 81 doesn\u0027t have a cube root."},{"Start":"01:49.555 ","End":"01:52.459","Text":"Well, it does, but not a whole number."},{"Start":"01:52.459 ","End":"01:55.790","Text":"There\u0027s a way of simplifying it a bit more."},{"Start":"01:55.790 ","End":"02:01.600","Text":"What I can do is find something that is a power of 3."},{"Start":"02:01.600 ","End":"02:09.080","Text":"Some cube that divides into 81 and we can see that 27 is the cube that goes into 81."},{"Start":"02:09.080 ","End":"02:13.270","Text":"In other words, the cube root of 27 times 3."},{"Start":"02:13.270 ","End":"02:16.415","Text":"Numbers like 27 should be very familiar to you."},{"Start":"02:16.415 ","End":"02:18.470","Text":"This is 3^3."},{"Start":"02:18.470 ","End":"02:21.230","Text":"I take the cube root of this product and then I do"},{"Start":"02:21.230 ","End":"02:24.200","Text":"this by taking the cube root of each piece separately."},{"Start":"02:24.200 ","End":"02:28.915","Text":"I\u0027ve got the cube root of 27 times the cube root of 3,"},{"Start":"02:28.915 ","End":"02:36.550","Text":"and the cube root of 27 is 3 because 3 times 3 times 3 is 27."},{"Start":"02:36.550 ","End":"02:43.220","Text":"This becomes 3 times cube root of 3."},{"Start":"02:43.220 ","End":"02:46.790","Text":"It\u0027s always debatable as to what it means to simplify."},{"Start":"02:46.790 ","End":"02:49.820","Text":"Is this simpler than the original?"},{"Start":"02:49.820 ","End":"02:53.885","Text":"If I wrote it, this part of 3^1/3 is that simpler?"},{"Start":"02:53.885 ","End":"02:56.720","Text":"Well, generally, it\u0027s accepted that this is what we do,"},{"Start":"02:56.720 ","End":"02:59.075","Text":"we write roots rather than"},{"Start":"02:59.075 ","End":"03:04.595","Text":"fractional exponents and if we can take out a whole number, this is what we do."},{"Start":"03:04.595 ","End":"03:08.855","Text":"This is considered simpler although aesthetically it\u0027s debatable."},{"Start":"03:08.855 ","End":"03:13.320","Text":"Number d minus 8^1/3."},{"Start":"03:13.320 ","End":"03:16.515","Text":"Notice that only the 8 is to the power of a 1/3."},{"Start":"03:16.515 ","End":"03:21.680","Text":"The minus is stuck in front because of order of operations exponents before subtraction."},{"Start":"03:21.680 ","End":"03:25.595","Text":"So it\u0027s minus and then we have the cube root,"},{"Start":"03:25.595 ","End":"03:27.662","Text":"that\u0027s what the power of a 1/3 means of 8."},{"Start":"03:27.662 ","End":"03:30.700","Text":"Cube root of 8 is 2,"},{"Start":"03:30.700 ","End":"03:32.660","Text":"so this is minus 2."},{"Start":"03:32.660 ","End":"03:37.635","Text":"Again why is the cube root of 8 to 2 because 2^3 equals 8."},{"Start":"03:37.635 ","End":"03:41.625","Text":"It\u0027s the opposite of cubing so that\u0027s that."},{"Start":"03:41.625 ","End":"03:47.580","Text":"Now, 32^1/5 is 5th root of 32."},{"Start":"03:47.580 ","End":"03:50.500","Text":"What number to the power of 5 is 32?"},{"Start":"03:50.500 ","End":"03:53.418","Text":"We just know these things from experience,"},{"Start":"03:53.418 ","End":"03:56.845","Text":"it\u0027s 2 because 2 to the 5th is 32,"},{"Start":"03:56.845 ","End":"04:01.274","Text":"2 times 2 times 2 times 2 times 2 doubling up to 4,"},{"Start":"04:01.274 ","End":"04:03.610","Text":"8, 16, 32."},{"Start":"04:03.650 ","End":"04:06.645","Text":"Let\u0027s see we have a few more to go,"},{"Start":"04:06.645 ","End":"04:09.460","Text":"this one minus 125^1/3."},{"Start":"04:10.280 ","End":"04:17.920","Text":"The minus is in front and then we have the cube root of 125."},{"Start":"04:17.920 ","End":"04:22.130","Text":"Once again, it\u0027s supposed to be familiar that 5 times 5 times 5 is"},{"Start":"04:22.130 ","End":"04:27.140","Text":"125 so that the cube root of 125 is 5."},{"Start":"04:27.140 ","End":"04:34.550","Text":"I\u0027ll just write at the side the reason that 5^3 is 125 means the cube root of 125 is 5."},{"Start":"04:34.550 ","End":"04:37.590","Text":"When we say cube, usually not to the power of 3"},{"Start":"04:37.590 ","End":"04:41.075","Text":"although t\u0027s possible 5^3, 5^3, whatever."},{"Start":"04:41.075 ","End":"04:44.040","Text":"I\u0027m old fashioned, I like to say cubed."},{"Start":"04:44.290 ","End":"04:51.330","Text":"16^3/4. This time we have in the fraction both the numerator and a denominator,"},{"Start":"04:51.330 ","End":"04:56.390","Text":"and there\u0027s actually two ways to go with this and you can choose which way to go."},{"Start":"04:56.390 ","End":"04:58.880","Text":"I\u0027ll demonstrate this here though in general,"},{"Start":"04:58.880 ","End":"05:00.760","Text":"I\u0027ll choose one or the other."},{"Start":"05:00.760 ","End":"05:03.190","Text":"The 3 means raising to the power of 3."},{"Start":"05:03.190 ","End":"05:04.800","Text":"The 4 means taking the 4th root."},{"Start":"05:04.800 ","End":"05:07.490","Text":"Thing is you can do these in either order you want."},{"Start":"05:07.490 ","End":"05:10.850","Text":"For example, you can take the 4th root of"},{"Start":"05:10.850 ","End":"05:15.740","Text":"16 and then raise this to the power of 3 and see what you get."},{"Start":"05:15.740 ","End":"05:18.650","Text":"Alternatively, you could first take 16 and raise"},{"Start":"05:18.650 ","End":"05:21.950","Text":"it to the power of 3 and then take the 4th root."},{"Start":"05:21.950 ","End":"05:24.260","Text":"Both will give you the same answer."},{"Start":"05:24.260 ","End":"05:26.240","Text":"You can do them in either order."},{"Start":"05:26.240 ","End":"05:29.990","Text":"I\u0027ll choose this one because I\u0027m dealing then with smaller numbers."},{"Start":"05:29.990 ","End":"05:34.340","Text":"I\u0027m not really sure what 16^3 is though probably not too hard to figure,"},{"Start":"05:34.340 ","End":"05:40.135","Text":"but the 4th root of 16 I know that this is 2 because 2^4 is 16,"},{"Start":"05:40.135 ","End":"05:41.970","Text":"2 times 2 times 2 times 2."},{"Start":"05:41.970 ","End":"05:44.550","Text":"So this is 2 and then cubed,"},{"Start":"05:44.550 ","End":"05:47.740","Text":"and then 2 times 2 times 2 is 8."},{"Start":"05:47.740 ","End":"05:49.985","Text":"This would have worked out."},{"Start":"05:49.985 ","End":"05:52.587","Text":"Let\u0027s see what would have happened if we did it the other way just for curiosity,"},{"Start":"05:52.587 ","End":"05:57.400","Text":"16^3, let\u0027s see,"},{"Start":"05:57.400 ","End":"06:02.670","Text":"binary numbers that in computers that is 2^4, 2^3,"},{"Start":"06:02.670 ","End":"06:09.075","Text":"2^12, I know that 2^10 is 1,024 so that makes it 4,096."},{"Start":"06:09.075 ","End":"06:12.840","Text":"Of course, you do it on a calculator and I didn\u0027t have one with me."},{"Start":"06:12.840 ","End":"06:15.060","Text":"Then we add the 4th root of that,"},{"Start":"06:15.060 ","End":"06:17.160","Text":"not really sure about that but you know what,"},{"Start":"06:17.160 ","End":"06:21.210","Text":"I\u0027ll just check that 8 to the 4th is 4,096 and that will justify it."},{"Start":"06:21.210 ","End":"06:23.235","Text":"Lets see, 8 times 8 is 64,"},{"Start":"06:23.235 ","End":"06:25.665","Text":"and 64 times 64,"},{"Start":"06:25.665 ","End":"06:28.620","Text":"you\u0027ll find that\u0027s 4,096."},{"Start":"06:28.620 ","End":"06:31.640","Text":"It was without a calculator but I\u0027m sure that if you do have a calculator,"},{"Start":"06:31.640 ","End":"06:34.490","Text":"check that the 4th root of this is this."},{"Start":"06:34.490 ","End":"06:36.200","Text":"Even if you don\u0027t have a 4th root function"},{"Start":"06:36.200 ","End":"06:37.340","Text":"you can take a square root and then square root."},{"Start":"06:37.340 ","End":"06:42.960","Text":"The square root of this should be 64. Looks about right."},{"Start":"06:42.960 ","End":"06:45.150","Text":"In general, if it\u0027s possible,"},{"Start":"06:45.150 ","End":"06:47.600","Text":"I\u0027ll take the root first and then the exponent."},{"Start":"06:47.600 ","End":"06:51.058","Text":"Like I said, I prefer this because then you\u0027re dealing with smaller numbers,"},{"Start":"06:51.058 ","End":"06:53.345","Text":"you see I got stuck with large numbers here."},{"Start":"06:53.345 ","End":"06:56.555","Text":"This one will do from experience with g,"},{"Start":"06:56.555 ","End":"07:00.000","Text":"will take the cube root of 27 first,"},{"Start":"07:00.000 ","End":"07:01.464","Text":"and then we\u0027ll raise that to the power of 2."},{"Start":"07:01.464 ","End":"07:03.990","Text":"Cube root of 27,"},{"Start":"07:03.990 ","End":"07:07.220","Text":"I have a feeling we\u0027ve done it somewhere further up and event that\u0027s"},{"Start":"07:07.220 ","End":"07:11.165","Text":"known to be 3 because 3^3 is 27."},{"Start":"07:11.165 ","End":"07:13.945","Text":"So this is 3,"},{"Start":"07:13.945 ","End":"07:17.390","Text":"the cube root part and then to the power of 2,"},{"Start":"07:17.390 ","End":"07:19.790","Text":"so this is 9."},{"Start":"07:19.790 ","End":"07:24.495","Text":"Last one, (243/32)^4/5."},{"Start":"07:24.495 ","End":"07:26.130","Text":"Because it\u0027s a fraction,"},{"Start":"07:26.130 ","End":"07:29.690","Text":"we can raise both numerator and denominator which is what we"},{"Start":"07:29.690 ","End":"07:33.320","Text":"have to do to the power of 4/5."},{"Start":"07:33.320 ","End":"07:42.900","Text":"So 243^4/5 over 32^4/5."},{"Start":"07:42.900 ","End":"07:44.510","Text":"Now what this is,"},{"Start":"07:44.510 ","End":"07:45.920","Text":"is that this is the 5th root of"},{"Start":"07:45.920 ","End":"07:55.175","Text":"243^4 over the 5th root"},{"Start":"07:55.175 ","End":"07:57.790","Text":"of 32 also to the power of 4."},{"Start":"07:57.790 ","End":"08:00.920","Text":"Like I said, there are two ways of doing it because I could"},{"Start":"08:00.920 ","End":"08:03.830","Text":"raise 32^4 first then taking the 5th root,"},{"Start":"08:03.830 ","End":"08:09.355","Text":"but when I don\u0027t have my calculator then I certainly prefer this route here."},{"Start":"08:09.355 ","End":"08:14.955","Text":"What we get is,"},{"Start":"08:14.955 ","End":"08:19.800","Text":"now this I know is 3 because I know"},{"Start":"08:19.800 ","End":"08:24.915","Text":"that we already had that 81 is 3^4 and 81 times 3 is this,"},{"Start":"08:24.915 ","End":"08:27.790","Text":"so this is 3^4."},{"Start":"08:27.860 ","End":"08:30.210","Text":"Once again to justify the 5th root,"},{"Start":"08:30.210 ","End":"08:36.540","Text":"3^5 is 243 as you can check, as I say,"},{"Start":"08:36.540 ","End":"08:41.220","Text":"3 times 3 times 3 times 3 already I know is 81 times 3 is 243,"},{"Start":"08:41.220 ","End":"08:45.120","Text":"and the 4 from here and this one 5th root of 32,"},{"Start":"08:45.120 ","End":"08:49.914","Text":"we\u0027ve done that before that\u0027s 2 or even because 2^5 is 32,"},{"Start":"08:49.914 ","End":"08:52.290","Text":"the power of 4, let\u0027s see now,"},{"Start":"08:52.290 ","End":"08:59.730","Text":"3^4 is 81, 2^4 is 16."},{"Start":"08:59.730 ","End":"09:02.090","Text":"Not clear if this is simple enough,"},{"Start":"09:02.090 ","End":"09:03.530","Text":"I would leave it like this,"},{"Start":"09:03.530 ","End":"09:04.950","Text":"but some might say, well,"},{"Start":"09:04.950 ","End":"09:09.073","Text":"it\u0027s an improper fraction so let\u0027s also write it as a mixed number,"},{"Start":"09:09.073 ","End":"09:12.240","Text":"16 into 81 goes 5 times."},{"Start":"09:12.920 ","End":"09:18.270","Text":"As I was saying, 5 times goes into 80 and there\u0027s one leftover,"},{"Start":"09:18.270 ","End":"09:20.310","Text":"so it\u0027s 5 and 116th."},{"Start":"09:20.310 ","End":"09:23.550","Text":"Although, I like improper fractions myself,"},{"Start":"09:23.550 ","End":"09:25.140","Text":"I think this looks simpler than this,"},{"Start":"09:25.140 ","End":"09:26.540","Text":"a question of aesthetics."},{"Start":"09:26.540 ","End":"09:30.960","Text":"Anyway, we\u0027re done with all 9 parts of this question."}],"ID":8109},{"Watched":false,"Name":"Exercise 6","Duration":"2m 30s","ChapterTopicVideoID":8017,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8017.jpeg","UploadDate":"2020-09-30T14:31:28.5900000","DurationForVideoObject":"PT2M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.645","Text":"This exercise has four separate parts."},{"Start":"00:03.645 ","End":"00:06.540","Text":"In each one we have radicals or roots."},{"Start":"00:06.540 ","End":"00:08.220","Text":"The word radical and root is the same thing."},{"Start":"00:08.220 ","End":"00:10.270","Text":"It means this thing here."},{"Start":"00:10.270 ","End":"00:12.990","Text":"We want to get rid of those and write it instead"},{"Start":"00:12.990 ","End":"00:16.965","Text":"with fractional exponents. The first one."},{"Start":"00:16.965 ","End":"00:18.870","Text":"When you have the nth root of something,"},{"Start":"00:18.870 ","End":"00:24.690","Text":"it means to the power of 1 over n. This is (16)^1\\4."},{"Start":"00:24.690 ","End":"00:28.350","Text":"This is basically what\u0027s expected, that\u0027s the answer."},{"Start":"00:28.350 ","End":"00:30.059","Text":"But because it\u0027s numerical,"},{"Start":"00:30.059 ","End":"00:32.040","Text":"I\u0027d like to actually also give"},{"Start":"00:32.040 ","End":"00:35.460","Text":"the final answer because it comes out a nice whole number 2,"},{"Start":"00:35.460 ","End":"00:38.130","Text":"the fourth root of 16 is also 2,"},{"Start":"00:38.130 ","End":"00:40.510","Text":"but I think this is what they were getting at."},{"Start":"00:40.510 ","End":"00:43.375","Text":"The fourth root, power of a 1/4."},{"Start":"00:43.375 ","End":"00:49.530","Text":"Second one what have here is the 10th root of 5x."},{"Start":"00:49.540 ","End":"00:58.455","Text":"What we can say is that this is equal to 5x to the power of 1 tenth."},{"Start":"00:58.455 ","End":"01:00.620","Text":"Really I would leave it at that."},{"Start":"01:00.620 ","End":"01:02.495","Text":"It doesn\u0027t say simplify,"},{"Start":"01:02.495 ","End":"01:07.504","Text":"but you could also write it as 5 to the power of a 10th,"},{"Start":"01:07.504 ","End":"01:09.935","Text":"x to the power of a 10th."},{"Start":"01:09.935 ","End":"01:13.820","Text":"This even has some numerical value that could be done on the calculator,"},{"Start":"01:13.820 ","End":"01:16.034","Text":"but I wouldn\u0027t do it."},{"Start":"01:16.034 ","End":"01:21.335","Text":"See, what we have here is the square root of something."},{"Start":"01:21.335 ","End":"01:25.850","Text":"We take that something and raise it to the power of a 1/2."},{"Start":"01:25.850 ","End":"01:28.160","Text":"Even for the question that says simplify,"},{"Start":"01:28.160 ","End":"01:29.960","Text":"there is no way to simplify,"},{"Start":"01:29.960 ","End":"01:34.700","Text":"avoid the temptation of taking each piece separately to the power of a 1/2."},{"Start":"01:34.700 ","End":"01:36.020","Text":"It works for multiplication,"},{"Start":"01:36.020 ","End":"01:38.045","Text":"but it does not work for addition."},{"Start":"01:38.045 ","End":"01:40.310","Text":"This is also a simplified as it gets,"},{"Start":"01:40.310 ","End":"01:43.075","Text":"then it\u0027s also fractional exponent."},{"Start":"01:43.075 ","End":"01:47.525","Text":"The last one, we have the cube root of a plus b to the 5th,"},{"Start":"01:47.525 ","End":"01:50.270","Text":"so it\u0027s a plus b."},{"Start":"01:50.270 ","End":"01:55.655","Text":"In one step, we can say it to the power of 5/3."},{"Start":"01:55.655 ","End":"02:00.230","Text":"Because whenever you have something in the root here,"},{"Start":"02:00.230 ","End":"02:02.765","Text":"the third root of something to the power of something."},{"Start":"02:02.765 ","End":"02:05.330","Text":"Maybe what I\u0027d like to say is that if I have"},{"Start":"02:05.330 ","End":"02:12.425","Text":"some quantity x and I take the nth root and the power of n,"},{"Start":"02:12.425 ","End":"02:16.790","Text":"and that\u0027s the same as x to the power of n over"},{"Start":"02:16.790 ","End":"02:21.740","Text":"m. Trying to sell it in words that this number here goes in the denominator,"},{"Start":"02:21.740 ","End":"02:24.590","Text":"this number here goes in the numerator."},{"Start":"02:24.590 ","End":"02:26.795","Text":"That\u0027s basically it."},{"Start":"02:26.795 ","End":"02:31.050","Text":"No more to it than that. We are done."}],"ID":8110},{"Watched":false,"Name":"Exercise 7","Duration":"8m 44s","ChapterTopicVideoID":8018,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8018.jpeg","UploadDate":"2020-09-30T14:34:28.5570000","DurationForVideoObject":"PT8M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.325","Text":"This exercise has several parts,"},{"Start":"00:02.325 ","End":"00:03.840","Text":"a through h,"},{"Start":"00:03.840 ","End":"00:07.215","Text":"in each one of them we have to simplify the expression."},{"Start":"00:07.215 ","End":"00:09.770","Text":"The reason we\u0027re going to assume that x, y,"},{"Start":"00:09.770 ","End":"00:13.150","Text":"and z, or z positive,"},{"Start":"00:13.150 ","End":"00:17.670","Text":"is so that we don\u0027t have to worry about square root of a negative number,"},{"Start":"00:17.670 ","End":"00:20.310","Text":"or any even order root of a negative number."},{"Start":"00:20.310 ","End":"00:22.605","Text":"Just not to worry about that."},{"Start":"00:22.605 ","End":"00:24.540","Text":"I did learn in England,"},{"Start":"00:24.540 ","End":"00:28.410","Text":"so if I sometimes say z instead of z,"},{"Start":"00:28.410 ","End":"00:30.120","Text":"I don\u0027t get confused."},{"Start":"00:30.120 ","End":"00:32.039","Text":"I know there\u0027s supposed to be for America,"},{"Start":"00:32.039 ","End":"00:34.725","Text":"and I should say z, anyway, never mind that."},{"Start":"00:34.725 ","End":"00:36.790","Text":"Let\u0027s start with Part a."},{"Start":"00:36.790 ","End":"00:41.120","Text":"We want the square root of x^6,"},{"Start":"00:41.120 ","End":"00:46.295","Text":"and what I suggest is writing the square root as a fractional exponent."},{"Start":"00:46.295 ","End":"00:51.860","Text":"We have x^6^1/2. How does this help us?"},{"Start":"00:51.860 ","End":"00:54.770","Text":"This is a fraction, because now I can use the rule of exponents and I can"},{"Start":"00:54.770 ","End":"00:58.040","Text":"multiply 6 times 1/2 and get 3."},{"Start":"00:58.040 ","End":"01:01.400","Text":"The answer is x^3 or x to the third."},{"Start":"01:01.400 ","End":"01:05.620","Text":"Next one, we have both power and the root."},{"Start":"01:05.620 ","End":"01:11.205","Text":"We right away can say that this is y^6/9."},{"Start":"01:11.205 ","End":"01:16.985","Text":"This is not the simplest answer because fractions should be reduced."},{"Start":"01:16.985 ","End":"01:22.055","Text":"So six-ninths, if you divide top and bottom by 3 is two-thirds."},{"Start":"01:22.055 ","End":"01:25.520","Text":"The answer is y^2/3."},{"Start":"01:25.520 ","End":"01:28.910","Text":"Actually it\u0027s better to leave the answer,"},{"Start":"01:28.910 ","End":"01:32.570","Text":"it\u0027s considered simpler to leave it in root or radical form,"},{"Start":"01:32.570 ","End":"01:35.240","Text":"so I\u0027ll write this as."},{"Start":"01:35.240 ","End":"01:36.920","Text":"What we do is rewrite the y."},{"Start":"01:36.920 ","End":"01:38.690","Text":"The thing at the top goes here,"},{"Start":"01:38.690 ","End":"01:40.790","Text":"the thing at the bottom goes here."},{"Start":"01:40.790 ","End":"01:43.160","Text":"That\u0027s just mechanically how we remember to do it."},{"Start":"01:43.160 ","End":"01:46.630","Text":"That\u0027s say the simplest."},{"Start":"01:46.630 ","End":"01:50.195","Text":"Next one, we have numbers,"},{"Start":"01:50.195 ","End":"01:52.850","Text":"and x\u0027s, and y\u0027s."},{"Start":"01:52.850 ","End":"01:57.520","Text":"What we do is apply this to each part separately."},{"Start":"01:57.520 ","End":"02:01.490","Text":"The first stage is to write square root of 18,"},{"Start":"02:01.490 ","End":"02:03.390","Text":"square root of x^3,"},{"Start":"02:03.390 ","End":"02:06.590","Text":"square root of y^7."},{"Start":"02:06.590 ","End":"02:12.365","Text":"The next step is we try and see,"},{"Start":"02:12.365 ","End":"02:14.150","Text":"since we\u0027re talking about square roots,"},{"Start":"02:14.150 ","End":"02:17.570","Text":"what\u0027s the largest square that will go into 18."},{"Start":"02:17.570 ","End":"02:21.080","Text":"I\u0027m going to write 18 as 9,"},{"Start":"02:21.080 ","End":"02:22.985","Text":"that\u0027s the largest square that goes into it."},{"Start":"02:22.985 ","End":"02:24.650","Text":"9 is a square number of course."},{"Start":"02:24.650 ","End":"02:27.515","Text":"I\u0027ll write it as 9 times 2."},{"Start":"02:27.515 ","End":"02:30.680","Text":"The x^3, I\u0027m going to rewrite something squared."},{"Start":"02:30.680 ","End":"02:33.175","Text":"Well x^2 is the best I can do,"},{"Start":"02:33.175 ","End":"02:36.130","Text":"and the other bit is x."},{"Start":"02:36.130 ","End":"02:38.710","Text":"Here, y^7,"},{"Start":"02:38.710 ","End":"02:40.895","Text":"I\u0027ll write it as y^6,"},{"Start":"02:40.895 ","End":"02:46.310","Text":"because this is an even number and now it\u0027s going to be something squared times y."},{"Start":"02:46.310 ","End":"02:49.760","Text":"Now each bit can be broken up."},{"Start":"02:49.760 ","End":"02:57.695","Text":"I can say that this equals the square root of 9 times the square root of 2."},{"Start":"02:57.695 ","End":"03:02.425","Text":"Here, the square root of x^2,"},{"Start":"03:02.425 ","End":"03:04.830","Text":"square root of x."},{"Start":"03:04.830 ","End":"03:07.935","Text":"Square root of y^6,"},{"Start":"03:07.935 ","End":"03:10.279","Text":"square root of y."},{"Start":"03:10.279 ","End":"03:13.130","Text":"Now the bits that were squares,"},{"Start":"03:13.130 ","End":"03:15.200","Text":"this one, this one,"},{"Start":"03:15.200 ","End":"03:18.020","Text":"and this one, can be simplified further"},{"Start":"03:18.020 ","End":"03:21.625","Text":"because the square root of 9 is a number that\u0027s 3."},{"Start":"03:21.625 ","End":"03:26.040","Text":"I have 3 times square root of 2."},{"Start":"03:26.040 ","End":"03:29.595","Text":"Now the square root of x^2 is just x,"},{"Start":"03:29.595 ","End":"03:33.560","Text":"and that\u0027s another reason why we assume that the variables are positive."},{"Start":"03:33.560 ","End":"03:41.390","Text":"Because the square root of x^2 in general is actually the absolute value of x."},{"Start":"03:41.390 ","End":"03:43.580","Text":"If it\u0027s positive, it\u0027s just x."},{"Start":"03:43.580 ","End":"03:44.960","Text":"But if it\u0027s negative,"},{"Start":"03:44.960 ","End":"03:46.955","Text":"we get the opposite sign."},{"Start":"03:46.955 ","End":"03:49.190","Text":"But because the x is positive,"},{"Start":"03:49.190 ","End":"03:51.725","Text":"then it is equal to just x."},{"Start":"03:51.725 ","End":"03:53.810","Text":"What I\u0027m saying is that I could write this"},{"Start":"03:53.810 ","End":"03:56.270","Text":"automatically because I knew that x was positive."},{"Start":"03:56.270 ","End":"03:59.765","Text":"I gave it some thought because normally square root of x^2 is tricky."},{"Start":"03:59.765 ","End":"04:01.835","Text":"Then square root of x,"},{"Start":"04:01.835 ","End":"04:03.960","Text":"square root of y^6."},{"Start":"04:03.960 ","End":"04:09.080","Text":"Mentally do that, is y^6/2, which is y^3."},{"Start":"04:09.080 ","End":"04:14.790","Text":"If you think about it, y^3 times y^3 is just y^6,"},{"Start":"04:14.790 ","End":"04:21.980","Text":"and then square root of y. I would think that this is the simplest way of writing it,"},{"Start":"04:21.980 ","End":"04:24.320","Text":"even though it doesn\u0027t look that simple."},{"Start":"04:24.320 ","End":"04:27.995","Text":"I don\u0027t see any point in putting the square roots separately or anything."},{"Start":"04:27.995 ","End":"04:31.880","Text":"I think this is simplified even if it\u0027s not simple."},{"Start":"04:31.880 ","End":"04:33.760","Text":"Best we can do."},{"Start":"04:33.760 ","End":"04:36.370","Text":"Onto the next. Here,"},{"Start":"04:36.370 ","End":"04:37.940","Text":"this is an automatic one,"},{"Start":"04:37.940 ","End":"04:40.595","Text":"when you have a number exponent here and a root,"},{"Start":"04:40.595 ","End":"04:43.150","Text":"it becomes a fractional exponent,"},{"Start":"04:43.150 ","End":"04:44.550","Text":"3 on the top,"},{"Start":"04:44.550 ","End":"04:46.115","Text":"5 on the bottom."},{"Start":"04:46.115 ","End":"04:49.265","Text":"That\u0027s it, a primitive operation."},{"Start":"04:49.265 ","End":"04:52.505","Text":"Next one, cube root of."},{"Start":"04:52.505 ","End":"04:55.870","Text":"We\u0027ll take the cube root of each piece separately,"},{"Start":"04:55.870 ","End":"05:02.165","Text":"and the cube root of something to the power of something is that power over 3."},{"Start":"05:02.165 ","End":"05:04.055","Text":"What I\u0027m saying is it\u0027ll take 2 steps in 1,"},{"Start":"05:04.055 ","End":"05:07.810","Text":"cube root of x^2 is x^2/3,"},{"Start":"05:07.810 ","End":"05:11.370","Text":"cube root of y cubed is y^3/3,"},{"Start":"05:11.370 ","End":"05:18.910","Text":"cube root of z^4 is z^4/3."},{"Start":"05:19.040 ","End":"05:21.350","Text":"Let\u0027s see what we can do with this,"},{"Start":"05:21.350 ","End":"05:23.480","Text":"if there\u0027s anything more that we can do."},{"Start":"05:23.480 ","End":"05:30.510","Text":"Yes, the fractional power could be written back as a radical."},{"Start":"05:30.510 ","End":"05:35.025","Text":"Here we have the cube root of x^2."},{"Start":"05:35.025 ","End":"05:37.980","Text":"Here, 3/3 happens to be 1,"},{"Start":"05:37.980 ","End":"05:40.995","Text":"so y^1 is just y."},{"Start":"05:40.995 ","End":"05:45.490","Text":"Here, what we could say is that 4/3,"},{"Start":"05:45.490 ","End":"05:46.970","Text":"I\u0027m talking fractions here,"},{"Start":"05:46.970 ","End":"05:49.205","Text":"as a mixed number is 1 1/3,"},{"Start":"05:49.205 ","End":"05:53.660","Text":"and 1 1/3 is actually 1 plus a third."},{"Start":"05:53.660 ","End":"05:57.890","Text":"I can write this as z^1,"},{"Start":"05:57.890 ","End":"06:00.020","Text":"which is just z, sorry,"},{"Start":"06:00.020 ","End":"06:01.700","Text":"z, if you\u0027re American."},{"Start":"06:01.700 ","End":"06:06.180","Text":"Z times z^1/3."},{"Start":"06:06.340 ","End":"06:10.880","Text":"Let me write that power of a third as cube root."},{"Start":"06:10.880 ","End":"06:20.095","Text":"I\u0027ll just erase it and write the z^1/3 as the cube root of z."},{"Start":"06:20.095 ","End":"06:24.280","Text":"This is about as simple as it\u0027ll get for simplification."},{"Start":"06:24.280 ","End":"06:27.295","Text":"Part f. What do we have here?"},{"Start":"06:27.295 ","End":"06:29.365","Text":"Again, we have a product,"},{"Start":"06:29.365 ","End":"06:30.940","Text":"and we have a fifth root,"},{"Start":"06:30.940 ","End":"06:33.325","Text":"so it\u0027s the fifth root of each."},{"Start":"06:33.325 ","End":"06:37.820","Text":"We have x^15/5,"},{"Start":"06:37.820 ","End":"06:47.090","Text":"then y^5/5, and z^20/5."},{"Start":"06:47.090 ","End":"06:53.140","Text":"You can also think of it from the beginning as this all to the power of a fifth,"},{"Start":"06:53.140 ","End":"06:55.340","Text":"and then each piece to the power of a fifth,"},{"Start":"06:55.340 ","End":"06:57.950","Text":"and you multiply exponents."},{"Start":"06:57.950 ","End":"07:02.420","Text":"These fractions come out nice because they will come out whole numbers,"},{"Start":"07:02.420 ","End":"07:05.420","Text":"15/5 is 3, so it\u0027s x^3."},{"Start":"07:05.420 ","End":"07:08.105","Text":"5/5 is 1,"},{"Start":"07:08.105 ","End":"07:09.995","Text":"y^1 is just y,"},{"Start":"07:09.995 ","End":"07:13.145","Text":"and 20/5 is 4,"},{"Start":"07:13.145 ","End":"07:17.285","Text":"so that makes it z^4."},{"Start":"07:17.285 ","End":"07:19.310","Text":"That\u0027s it. This is simple,"},{"Start":"07:19.310 ","End":"07:21.110","Text":"much simpler than this anyway."},{"Start":"07:21.110 ","End":"07:23.945","Text":"Part g, we have a cube root here,"},{"Start":"07:23.945 ","End":"07:25.835","Text":"very similar to this."},{"Start":"07:25.835 ","End":"07:31.280","Text":"It\u0027s x^3/3, y^6/3."},{"Start":"07:31.280 ","End":"07:35.495","Text":"This thing just becomes the denominator of the fractions here."},{"Start":"07:35.495 ","End":"07:39.710","Text":"Once again, the fractions come out whole numbers,"},{"Start":"07:39.710 ","End":"07:42.900","Text":"so it\u0027s just xy^2."},{"Start":"07:42.970 ","End":"07:46.010","Text":"Last but not least,"},{"Start":"07:46.010 ","End":"07:48.620","Text":"we have a product of 2 things here."},{"Start":"07:48.620 ","End":"07:52.520","Text":"The first one to the power of the fifth root,"},{"Start":"07:52.520 ","End":"07:54.230","Text":"therefore it\u0027s the power of a fifth."},{"Start":"07:54.230 ","End":"08:02.070","Text":"We get x^15/5, y^5/5."},{"Start":"08:02.070 ","End":"08:03.260","Text":"Then we just keep going."},{"Start":"08:03.260 ","End":"08:07.835","Text":"It\u0027s also a fifth root, and z^20/5."},{"Start":"08:07.835 ","End":"08:12.590","Text":"Would have been the same if I had a fifth root over the whole thing actually."},{"Start":"08:12.590 ","End":"08:18.290","Text":"Anyway. Once again, this comes out to be a whole number. They all do."},{"Start":"08:18.290 ","End":"08:22.820","Text":"x^3, y^1,"},{"Start":"08:22.820 ","End":"08:28.035","Text":"which is just y, and z^4."},{"Start":"08:28.035 ","End":"08:31.020","Text":"Kept thinking, \"This is familiar.\""},{"Start":"08:31.020 ","End":"08:32.635","Text":"Look at Part f,"},{"Start":"08:32.635 ","End":"08:35.840","Text":"the same thing, except that this was split up."},{"Start":"08:35.840 ","End":"08:37.940","Text":"Here is 2 separate fifth roots."},{"Start":"08:37.940 ","End":"08:39.860","Text":"That\u0027s what I was saying earlier."},{"Start":"08:39.860 ","End":"08:45.120","Text":"Of course we get the same answer. We\u0027re done."}],"ID":8111},{"Watched":false,"Name":"Exercise 8","Duration":"5m 16s","ChapterTopicVideoID":8019,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8019.jpeg","UploadDate":"2020-09-30T14:36:35.1900000","DurationForVideoObject":"PT5M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"Here we have a question in 3 parts."},{"Start":"00:03.060 ","End":"00:06.435","Text":"We have to multiply binomial,"},{"Start":"00:06.435 ","End":"00:08.520","Text":"I mean something with 2 terms by something with"},{"Start":"00:08.520 ","End":"00:13.025","Text":"2 terms in each case and we have square roots."},{"Start":"00:13.025 ","End":"00:15.474","Text":"Let\u0027s start with the first."},{"Start":"00:15.474 ","End":"00:17.130","Text":"When we have 2 things times 2 things,"},{"Start":"00:17.130 ","End":"00:20.475","Text":"you multiply each thing in here with each thing in here,"},{"Start":"00:20.475 ","End":"00:23.300","Text":"like this with this, this with this,"},{"Start":"00:23.300 ","End":"00:26.340","Text":"this with this, this with this doesn\u0027t matter what order."},{"Start":"00:26.340 ","End":"00:29.700","Text":"I\u0027m just reminding you that\u0027s how we handle this kind of thing."},{"Start":"00:29.700 ","End":"00:33.020","Text":"I also want to remind you that in general,"},{"Start":"00:33.020 ","End":"00:37.010","Text":"if we have the square root of a times the square root of a,"},{"Start":"00:37.010 ","End":"00:41.660","Text":"then this is equal to a where a could be anything could be x,"},{"Start":"00:41.660 ","End":"00:44.210","Text":"could be y than the last one."},{"Start":"00:44.210 ","End":"00:48.470","Text":"Of course, in these exercises we are going to assume that x is positive."},{"Start":"00:48.470 ","End":"00:53.990","Text":"I guess we\u0027re also going to assume I forgot to write that y2 is positive,"},{"Start":"00:53.990 ","End":"00:58.580","Text":"we don\u0027t have to worry about square roots of negative numbers."},{"Start":"00:58.580 ","End":"01:00.185","Text":"In the first one,"},{"Start":"01:00.185 ","End":"01:05.465","Text":"what we have is this with this root x times root x is just x."},{"Start":"01:05.465 ","End":"01:08.695","Text":"Let\u0027s take next this, with this."},{"Start":"01:08.695 ","End":"01:12.875","Text":"It\u0027s minus 2 times root x."},{"Start":"01:12.875 ","End":"01:14.720","Text":"Put the number before the x part,"},{"Start":"01:14.720 ","End":"01:16.910","Text":"I wouldn\u0027t write root x2,"},{"Start":"01:16.910 ","End":"01:22.235","Text":"2 goes in front and here plus 2 root x."},{"Start":"01:22.235 ","End":"01:32.135","Text":"Then 2 with minus 2 is minus 4 and the middle bit cancels minus 2 root x with 2 root x."},{"Start":"01:32.135 ","End":"01:34.745","Text":"We are left with x minus 4."},{"Start":"01:34.745 ","End":"01:37.325","Text":"Of course, this could be done better."},{"Start":"01:37.325 ","End":"01:42.424","Text":"If you remembered that there is something called a difference of squares formula,"},{"Start":"01:42.424 ","End":"01:50.270","Text":"which I write as a plus b times a minus b is a^2 minus b^2."},{"Start":"01:50.270 ","End":"01:51.979","Text":"If you remember this,"},{"Start":"01:51.979 ","End":"01:54.320","Text":"then it would have gone easier."},{"Start":"01:54.320 ","End":"01:56.660","Text":"I\u0027ll do it in another way also,"},{"Start":"01:56.660 ","End":"01:58.460","Text":"this is fine. This is the answer."},{"Start":"01:58.460 ","End":"02:06.690","Text":"But the alternative way of doing it is to say this is the square root of x^2 minus 2^2."},{"Start":"02:06.690 ","End":"02:12.050","Text":"Then square root of x times square root of x is just x and 2 times 2 is 4."},{"Start":"02:12.050 ","End":"02:14.795","Text":"We get x minus 4 same thing."},{"Start":"02:14.795 ","End":"02:16.550","Text":"Really this is the way to go,"},{"Start":"02:16.550 ","End":"02:18.680","Text":"but you might not have remembered the formula."},{"Start":"02:18.680 ","End":"02:20.840","Text":"Now onto part b."},{"Start":"02:20.840 ","End":"02:23.090","Text":"Part b, you can\u0027t use a difference of squares"},{"Start":"02:23.090 ","End":"02:25.700","Text":"because it has to be plus and minus the same thing."},{"Start":"02:25.700 ","End":"02:30.460","Text":"This time we really do have to multiply each in here with each in here."},{"Start":"02:30.460 ","End":"02:32.280","Text":"I\u0027ll put the arcs in it,"},{"Start":"02:32.280 ","End":"02:33.720","Text":"sometimes helps this with this,"},{"Start":"02:33.720 ","End":"02:36.630","Text":"this with this, this with this, this with this."},{"Start":"02:36.630 ","End":"02:39.825","Text":"We get this with this is x,"},{"Start":"02:39.825 ","End":"02:46.847","Text":"this with this twice square root of x and then minus 5 times root x."},{"Start":"02:46.847 ","End":"02:48.380","Text":"Remember write the number in front."},{"Start":"02:48.380 ","End":"02:50.830","Text":"Don\u0027t write root x times 5,"},{"Start":"02:50.830 ","End":"02:53.955","Text":"just looks better, it\u0027s easier, less confusing."},{"Start":"02:53.955 ","End":"02:58.265","Text":"2 with minus 5 is minus 10 and there are"},{"Start":"02:58.265 ","End":"03:03.130","Text":"things to be combined because root x is common here and here."},{"Start":"03:03.130 ","End":"03:12.420","Text":"I can say 2 minus 5 is minus 3 and write it as x minus 3 root x minus 10."},{"Start":"03:12.420 ","End":"03:14.835","Text":"In the last one again,"},{"Start":"03:14.835 ","End":"03:17.360","Text":"we have 4 multiplications."},{"Start":"03:17.360 ","End":"03:19.490","Text":"This times this, this times this,"},{"Start":"03:19.490 ","End":"03:21.965","Text":"this times this, this times this."},{"Start":"03:21.965 ","End":"03:26.155","Text":"Add them altogether or subtract as the case may be."},{"Start":"03:26.155 ","End":"03:29.550","Text":"4 root x times 2 root x."},{"Start":"03:29.550 ","End":"03:31.485","Text":"This one I\u0027ll just do at the side,"},{"Start":"03:31.485 ","End":"03:35.550","Text":"4 root x times 2 root x."},{"Start":"03:35.550 ","End":"03:37.830","Text":"Order of multiplication makes no difference."},{"Start":"03:37.830 ","End":"03:42.855","Text":"I can take first the 4 with the 2 and then the root x with the root x."},{"Start":"03:42.855 ","End":"03:49.950","Text":"This gives 8 times x because 4 times 2 is 8 and root x and root x is x."},{"Start":"03:49.950 ","End":"03:52.400","Text":"Then here, this times this,"},{"Start":"03:52.400 ","End":"03:54.125","Text":"I\u0027ll just put the number first."},{"Start":"03:54.125 ","End":"03:56.722","Text":"It\u0027s twice. You know what?"},{"Start":"03:56.722 ","End":"03:57.995","Text":"Let\u0027s put it in order."},{"Start":"03:57.995 ","End":"04:00.455","Text":"I\u0027ll put the root x before the root y."},{"Start":"04:00.455 ","End":"04:02.315","Text":"If I didn\u0027t do that,"},{"Start":"04:02.315 ","End":"04:04.505","Text":"you might not see, well,"},{"Start":"04:04.505 ","End":"04:06.710","Text":"let me write the next term on and you\u0027ll see my point."},{"Start":"04:06.710 ","End":"04:10.130","Text":"Next term would be this with this."},{"Start":"04:10.130 ","End":"04:12.905","Text":"Once again do the numbers first,"},{"Start":"04:12.905 ","End":"04:17.795","Text":"it\u0027s 20, and then the root x, and then the root y."},{"Start":"04:17.795 ","End":"04:21.455","Text":"Going back here, if I had written root y root x,"},{"Start":"04:21.455 ","End":"04:26.685","Text":"I might not have seen that these 2 are actually the same thing."},{"Start":"04:26.685 ","End":"04:29.480","Text":"Maybe I wouldn\u0027t have combined like terms,"},{"Start":"04:29.480 ","End":"04:31.400","Text":"which I\u0027m about to do anyway,"},{"Start":"04:31.400 ","End":"04:33.980","Text":"let\u0027s just put the last of the 4 in here."},{"Start":"04:33.980 ","End":"04:41.445","Text":"This times this first the 5 minus and then root y times root y is y, it\u0027s 5y."},{"Start":"04:41.445 ","End":"04:45.740","Text":"But because of this root y root x being equal to root x root y I"},{"Start":"04:45.740 ","End":"04:50.285","Text":"do have common similar terms to combine."},{"Start":"04:50.285 ","End":"04:59.525","Text":"I can say 8x plus 2 minus 20 of these is minus 18 of these root x root y,"},{"Start":"04:59.525 ","End":"05:02.450","Text":"and then minus 5y."},{"Start":"05:02.450 ","End":"05:05.390","Text":"Just pointing this out because you could\u0027ve"},{"Start":"05:05.390 ","End":"05:07.790","Text":"missed it and then you wouldn\u0027t have simplified it all the way."},{"Start":"05:07.790 ","End":"05:13.250","Text":"You would have left it as 2 root y root x minus 20."},{"Start":"05:13.250 ","End":"05:15.960","Text":"We are done here."}],"ID":8112},{"Watched":false,"Name":"Exercise 9","Duration":"9m 16s","ChapterTopicVideoID":8020,"CourseChapterTopicPlaylistID":56155,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8020.jpeg","UploadDate":"2020-09-30T14:39:37.8470000","DurationForVideoObject":"PT9M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"This exercise has 6 parts I see,"},{"Start":"00:03.270 ","End":"00:04.830","Text":"and in each part,"},{"Start":"00:04.830 ","End":"00:09.615","Text":"what you have to do is to rationalize the denominator."},{"Start":"00:09.615 ","End":"00:11.970","Text":"What this means is that you need to"},{"Start":"00:11.970 ","End":"00:17.295","Text":"algebraically manipulate it so that there are no roots in the denominator,"},{"Start":"00:17.295 ","End":"00:19.890","Text":"only possibly in the numerator."},{"Start":"00:19.890 ","End":"00:23.895","Text":"That\u0027s the term what it means rationalize the denominator."},{"Start":"00:23.895 ","End":"00:26.535","Text":"Let\u0027s begin with part a."},{"Start":"00:26.535 ","End":"00:29.070","Text":"We are assuming x is positive,"},{"Start":"00:29.070 ","End":"00:34.200","Text":"so we don\u0027t have questions like the square root of a negative number or something."},{"Start":"00:34.200 ","End":"00:38.985","Text":"Part a, we have 5 over the square root of x,"},{"Start":"00:38.985 ","End":"00:42.415","Text":"and there\u0027s a standard trick that one uses."},{"Start":"00:42.415 ","End":"00:51.545","Text":"What you do is you multiply top and bottom by this square root."},{"Start":"00:51.545 ","End":"00:54.014","Text":"The square root that\u0027s in the denominator,"},{"Start":"00:54.014 ","End":"00:56.030","Text":"just multiply top and bottom."},{"Start":"00:56.030 ","End":"00:59.300","Text":"This is okay because this fraction equals 1,"},{"Start":"00:59.300 ","End":"01:00.725","Text":"something over itself,"},{"Start":"01:00.725 ","End":"01:03.095","Text":"and x is positive so it\u0027s not 0,"},{"Start":"01:03.095 ","End":"01:06.560","Text":"so we don\u0027t have to worry about 0 in the denominator or anything."},{"Start":"01:06.560 ","End":"01:11.940","Text":"What we get is multiply the tops, 5 root x,"},{"Start":"01:11.940 ","End":"01:13.470","Text":"multiply the bottoms,"},{"Start":"01:13.470 ","End":"01:17.670","Text":"root x times root x is x. Yeah,"},{"Start":"01:17.670 ","End":"01:19.940","Text":"this could be left as an answer."},{"Start":"01:19.940 ","End":"01:24.110","Text":"We\u0027ve rationalized the denominator. Next one."},{"Start":"01:24.110 ","End":"01:27.440","Text":"This time, it isn\u0027t a square root,"},{"Start":"01:27.440 ","End":"01:29.165","Text":"it\u0027s a cube root."},{"Start":"01:29.165 ","End":"01:32.270","Text":"Cube root, it works differently."},{"Start":"01:32.270 ","End":"01:35.570","Text":"Cube root of x is x^1/3."},{"Start":"01:35.570 ","End":"01:38.900","Text":"That\u0027s what cube root of x means."},{"Start":"01:38.900 ","End":"01:44.000","Text":"In order to get this to be without roots,"},{"Start":"01:44.000 ","End":"01:49.670","Text":"I\u0027m going to multiply it by x^2/3."},{"Start":"01:49.670 ","End":"01:52.205","Text":"X^2/3 if I multiplied by this,"},{"Start":"01:52.205 ","End":"01:56.510","Text":"then it will give me 1/3 plus 2/3 is 1 and there won\u0027t be any roots."},{"Start":"01:56.510 ","End":"02:01.893","Text":"What we\u0027re going to do is multiply top and bottom. You know what?"},{"Start":"02:01.893 ","End":"02:05.430","Text":"First of all, just to use this technique,"},{"Start":"02:05.430 ","End":"02:08.310","Text":"I\u0027ll write it as 2 over x^1/3,"},{"Start":"02:08.310 ","End":"02:11.595","Text":"would be easier to explain."},{"Start":"02:11.595 ","End":"02:15.090","Text":"Now we\u0027re going to multiply by what I said,"},{"Start":"02:15.090 ","End":"02:17.900","Text":"x^2/3 on the bottom,"},{"Start":"02:17.900 ","End":"02:22.655","Text":"but of course also on the top because I can\u0027t go and just multiply the bottom."},{"Start":"02:22.655 ","End":"02:26.765","Text":"Then I multiply the fraction by 1 and that\u0027s okay."},{"Start":"02:26.765 ","End":"02:34.645","Text":"What I get on the denominator now is x^1/3 plus 2/3, which is 1."},{"Start":"02:34.645 ","End":"02:36.720","Text":"This becomes just x,"},{"Start":"02:36.720 ","End":"02:39.910","Text":"and on the top 2x^2/3."},{"Start":"02:40.390 ","End":"02:44.480","Text":"Now, because the original question was written with"},{"Start":"02:44.480 ","End":"02:48.095","Text":"roots and not with fractional exponents,"},{"Start":"02:48.095 ","End":"02:50.719","Text":"I\u0027ll also do the same thing with the answer."},{"Start":"02:50.719 ","End":"02:52.445","Text":"It\u0027s the right thing to do."},{"Start":"02:52.445 ","End":"02:55.475","Text":"Give it in the original terms."},{"Start":"02:55.475 ","End":"02:57.425","Text":"I\u0027ll write it as twice,"},{"Start":"02:57.425 ","End":"03:03.660","Text":"and then x^2/3 is the cube root of x^2 over x."},{"Start":"03:03.660 ","End":"03:09.065","Text":"Now I have no roots in the denominator and this expression is equal to this expression."},{"Start":"03:09.065 ","End":"03:11.980","Text":"In part c,"},{"Start":"03:11.980 ","End":"03:17.555","Text":"we\u0027ll also use fractional exponents for the middle stages."},{"Start":"03:17.555 ","End":"03:20.795","Text":"Fourth root means to the power of a quarter."},{"Start":"03:20.795 ","End":"03:22.610","Text":"If it\u0027s still the power of a quarter,"},{"Start":"03:22.610 ","End":"03:26.300","Text":"I\u0027m going to apply it to the top and the bottom separately,"},{"Start":"03:26.300 ","End":"03:32.540","Text":"so it\u0027s 2^1/4 over x^3/4."},{"Start":"03:32.540 ","End":"03:36.875","Text":"I\u0027ve skipped a couple of steps here because we\u0027ve done this thing so often."},{"Start":"03:36.875 ","End":"03:39.125","Text":"You can see that this is so,"},{"Start":"03:39.125 ","End":"03:41.120","Text":"and now to rationalize it,"},{"Start":"03:41.120 ","End":"03:43.405","Text":"I want to get it to be a whole number."},{"Start":"03:43.405 ","End":"03:48.710","Text":"So the thing to multiply the denominator by is x^1/4,"},{"Start":"03:48.710 ","End":"03:51.905","Text":"because then I\u0027ll get 3/4 plus a 1/4 is 1."},{"Start":"03:51.905 ","End":"03:54.770","Text":"Of course, if I do this to the denominator,"},{"Start":"03:54.770 ","End":"03:57.290","Text":"I must do it also to the numerator."},{"Start":"03:57.290 ","End":"04:00.195","Text":"This fraction is equal to 1, we are okay."},{"Start":"04:00.195 ","End":"04:08.790","Text":"What we get is 2^1/4, x^1/4 over x^1."},{"Start":"04:08.790 ","End":"04:15.050","Text":"Like I said, we should convert back from fractional exponents to roots or radicals,"},{"Start":"04:15.050 ","End":"04:17.645","Text":"because that\u0027s how it was given."},{"Start":"04:17.645 ","End":"04:19.835","Text":"This is equal to,"},{"Start":"04:19.835 ","End":"04:23.525","Text":"I\u0027ll write this as the fourth root of 2,"},{"Start":"04:23.525 ","End":"04:28.310","Text":"which is some number on the calculator can give you an approximation,"},{"Start":"04:28.310 ","End":"04:29.875","Text":"just leave it like this,"},{"Start":"04:29.875 ","End":"04:35.445","Text":"and then the fourth root of x over x."},{"Start":"04:35.445 ","End":"04:37.280","Text":"We\u0027ve rationalized the denominator,"},{"Start":"04:37.280 ","End":"04:39.140","Text":"no roots in the denominator."},{"Start":"04:39.140 ","End":"04:40.565","Text":"Let\u0027s go on to the next,"},{"Start":"04:40.565 ","End":"04:43.400","Text":"see if we can get the last 3 of them all in here."},{"Start":"04:43.400 ","End":"04:44.690","Text":"Yes, we have d, e,"},{"Start":"04:44.690 ","End":"04:46.170","Text":"and f. Now,"},{"Start":"04:46.170 ","End":"04:49.190","Text":"in this case, there\u0027s a different technique to be used."},{"Start":"04:49.190 ","End":"04:51.890","Text":"I hope you remember what a conjugate is,"},{"Start":"04:51.890 ","End":"04:54.626","Text":"but if not I\u0027ll give you a brief reminder."},{"Start":"04:54.626 ","End":"05:01.640","Text":"When you have a plus or minus of 2 things and 1 or both of them has a square root,"},{"Start":"05:01.640 ","End":"05:05.510","Text":"then you multiply by the thing that\u0027s similar to this,"},{"Start":"05:05.510 ","End":"05:08.280","Text":"just same but with the opposite sign."},{"Start":"05:08.280 ","End":"05:10.625","Text":"If it\u0027s minus then plus and vice versa."},{"Start":"05:10.625 ","End":"05:16.085","Text":"In other words here what I have to multiply the denominator by is its conjugate,"},{"Start":"05:16.085 ","End":"05:19.455","Text":"which is the square root of x plus 1."},{"Start":"05:19.455 ","End":"05:21.780","Text":"Same thing with a different sign,"},{"Start":"05:21.780 ","End":"05:26.330","Text":"and of course the numerator also because we have to balance things out."},{"Start":"05:26.330 ","End":"05:29.750","Text":"Now, let\u0027s see if we multiply what does this become?"},{"Start":"05:29.750 ","End":"05:32.233","Text":"I\u0027m going to just write the formula at the side,"},{"Start":"05:32.233 ","End":"05:40.710","Text":"the famous difference of squares formula that a^2 minus b^2 is equal to a minus b,"},{"Start":"05:40.710 ","End":"05:44.025","Text":"a plus b, or vice versa."},{"Start":"05:44.025 ","End":"05:46.760","Text":"I guess I\u0027m going to use it in this direction."},{"Start":"05:46.760 ","End":"05:49.370","Text":"This will be my a and this is the b,"},{"Start":"05:49.370 ","End":"05:51.740","Text":"so a minus b, a plus b."},{"Start":"05:51.740 ","End":"05:55.160","Text":"What is a^2 if a is square root of x?"},{"Start":"05:55.160 ","End":"05:57.570","Text":"That is just x."},{"Start":"05:57.570 ","End":"05:59.386","Text":"What is 1^2?"},{"Start":"05:59.386 ","End":"06:00.765","Text":"It\u0027s just 1,"},{"Start":"06:00.765 ","End":"06:02.280","Text":"but there\u0027s a minus."},{"Start":"06:02.280 ","End":"06:08.790","Text":"In the numerator just multiply out as usual,"},{"Start":"06:08.790 ","End":"06:14.600","Text":"in fact I could even just leave it as 5 times the square root of x plus 1."},{"Start":"06:14.600 ","End":"06:19.340","Text":"We don\u0027t need to simplify and I\u0027m not sure this is not simpler anyway."},{"Start":"06:19.340 ","End":"06:21.770","Text":"Certainly, we\u0027ve rationalized the denominator."},{"Start":"06:21.770 ","End":"06:23.720","Text":"There are no roots in the denominator."},{"Start":"06:23.720 ","End":"06:26.645","Text":"I\u0027ll leave it like that. Now the next one,"},{"Start":"06:26.645 ","End":"06:32.375","Text":"same idea, multiply top and bottom by the conjugate of the denominator."},{"Start":"06:32.375 ","End":"06:34.670","Text":"Don\u0027t worry about the word conjugate,"},{"Start":"06:34.670 ","End":"06:37.205","Text":"I don\u0027t like it myself, too technical."},{"Start":"06:37.205 ","End":"06:39.055","Text":"You have a root here,"},{"Start":"06:39.055 ","End":"06:42.935","Text":"and it could be in both as in the following question."},{"Start":"06:42.935 ","End":"06:47.735","Text":"When you have plus or minus sum or difference of something with a root,"},{"Start":"06:47.735 ","End":"06:52.449","Text":"you just take the opposite sign if it\u0027s a subtraction and then make it an addition."},{"Start":"06:52.449 ","End":"06:55.050","Text":"If it\u0027s an addition you make it a subtraction."},{"Start":"06:55.050 ","End":"06:59.265","Text":"Here we\u0027ll have 3 square root of x plus 4,"},{"Start":"06:59.265 ","End":"07:00.800","Text":"and the same thing on the top."},{"Start":"07:00.800 ","End":"07:02.390","Text":"That\u0027s the one thing that\u0027s very important."},{"Start":"07:02.390 ","End":"07:04.445","Text":"In other words, you\u0027ve changed the exercise."},{"Start":"07:04.445 ","End":"07:07.135","Text":"I\u0027m going to use this thing again,"},{"Start":"07:07.135 ","End":"07:10.670","Text":"a minus b, a plus b is a^2 minus b^2."},{"Start":"07:10.670 ","End":"07:13.670","Text":"This time this is the a and this is the b,"},{"Start":"07:13.670 ","End":"07:19.100","Text":"and so we get 3 root x times 3 root x"},{"Start":"07:19.100 ","End":"07:27.165","Text":"is 9x because it\u0027s 3 times 3 is 9 and root x times root x is x."},{"Start":"07:27.165 ","End":"07:33.840","Text":"Then, the minus b^2 is minus 16 because 4^2 is 16."},{"Start":"07:33.840 ","End":"07:42.440","Text":"On the numerator, I can leave it as twice 3 root x plus 4. That\u0027s fine."},{"Start":"07:42.440 ","End":"07:52.040","Text":"If you want to write the numerator as 6 root x plus 8, that\u0027s also possible."},{"Start":"07:52.040 ","End":"07:57.590","Text":"Both are okay. That\u0027s part e. We\u0027ve rationalized the denominator here,"},{"Start":"07:57.590 ","End":"07:59.520","Text":"we have one more to do."},{"Start":"07:59.520 ","End":"08:03.920","Text":"This time it\u0027s slightly different in the sense that they both have square roots in them,"},{"Start":"08:03.920 ","End":"08:05.490","Text":"but it\u0027s still the same technique."},{"Start":"08:05.490 ","End":"08:07.970","Text":"Like I said, if it\u0027s one or both with a square root,"},{"Start":"08:07.970 ","End":"08:10.595","Text":"we still use this concept of a conjugate,"},{"Start":"08:10.595 ","End":"08:16.160","Text":"which is the same thing as this almost 2 root x and root 5,"},{"Start":"08:16.160 ","End":"08:21.995","Text":"but change the sign of the operation if it was plus to minus and vice versa,"},{"Start":"08:21.995 ","End":"08:25.790","Text":"and we multiply numerator by the same thing of course."},{"Start":"08:25.790 ","End":"08:29.330","Text":"What we get using this formula again,"},{"Start":"08:29.330 ","End":"08:32.300","Text":"could have been plus and minus instead of minus and plus."},{"Start":"08:32.300 ","End":"08:36.365","Text":"Of course that makes no difference so we need the a^2 minus b^2."},{"Start":"08:36.365 ","End":"08:39.910","Text":"2 root x^2 is 4x,"},{"Start":"08:39.910 ","End":"08:43.405","Text":"because 2^2 is 4 root x^2 is x,"},{"Start":"08:43.405 ","End":"08:48.745","Text":"and then root 5^2 is 5."},{"Start":"08:48.745 ","End":"08:53.045","Text":"What we end up on the denominator is 4x minus 5."},{"Start":"08:53.045 ","End":"08:54.740","Text":"It\u0027s already rationalized."},{"Start":"08:54.740 ","End":"08:57.120","Text":"The numerator, well like I said,"},{"Start":"08:57.120 ","End":"09:00.980","Text":"you could leave the 5 outside or you could expand"},{"Start":"09:00.980 ","End":"09:05.870","Text":"the brackets and write it as just like we did above."},{"Start":"09:05.870 ","End":"09:09.180","Text":"I could write it say 10 root x minus 5 root 5."},{"Start":"09:09.180 ","End":"09:10.505","Text":"I\u0027ll leave it like this."},{"Start":"09:10.505 ","End":"09:16.200","Text":"That\u0027s fine. Do believe this is the last one so we\u0027re done."}],"ID":8113}],"Thumbnail":null,"ID":56155},{"Name":"Exponential Equations (Like Bases)","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"intro","Duration":"9m 20s","ChapterTopicVideoID":8022,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8022.jpeg","UploadDate":"2020-09-30T14:11:21.6300000","DurationForVideoObject":"PT9M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.010","Text":"In this section, we\u0027re going start talking about exponential equations."},{"Start":"00:05.010 ","End":"00:08.880","Text":"The first thing I\u0027ll do is show you what 1 might look like."},{"Start":"00:08.880 ","End":"00:18.825","Text":"Three to the power of 5x minus 3=3 to the power of 3x plus 7."},{"Start":"00:18.825 ","End":"00:21.555","Text":"What makes this an exponential equation?"},{"Start":"00:21.555 ","End":"00:24.885","Text":"Our variable, which is clearly x,"},{"Start":"00:24.885 ","End":"00:29.070","Text":"appears in the power or exponent of the equation."},{"Start":"00:29.070 ","End":"00:33.795","Text":"X appears in the exponent and they also x appears as part of the exponent."},{"Start":"00:33.795 ","End":"00:36.690","Text":"This is what makes it an exponential equation."},{"Start":"00:36.690 ","End":"00:39.555","Text":"In a moment, I\u0027m going to write down our first rule or"},{"Start":"00:39.555 ","End":"00:43.625","Text":"tool for solving exponential equations of certain kinds."},{"Start":"00:43.625 ","End":"00:47.420","Text":"But let me just point out here than this 1 that we have an exponent"},{"Start":"00:47.420 ","End":"00:51.340","Text":"on the left and an exponent on the right and it is the same base,"},{"Start":"00:51.340 ","End":"00:53.455","Text":"3 here, 3 here."},{"Start":"00:53.455 ","End":"00:55.520","Text":"This is the case we\u0027re going to focus on,"},{"Start":"00:55.520 ","End":"00:56.720","Text":"not necessarily 3,"},{"Start":"00:56.720 ","End":"00:58.790","Text":"but same base on both sides."},{"Start":"00:58.790 ","End":"01:02.750","Text":"It turns out that if you have exponents with the same base,"},{"Start":"01:02.750 ","End":"01:06.845","Text":"the way to solve it is just to throw out the base if you like,"},{"Start":"01:06.845 ","End":"01:09.575","Text":"or in other words compare the exponents."},{"Start":"01:09.575 ","End":"01:16.685","Text":"In this case, I would say that 5x minus 3=3x plus 7."},{"Start":"01:16.685 ","End":"01:19.595","Text":"But let me just write something down more formal."},{"Start":"01:19.595 ","End":"01:24.650","Text":"That is if we have an equation of the form a to the power of 1 thing,"},{"Start":"01:24.650 ","End":"01:29.000","Text":"I\u0027ll just call it square equals a to the power of another thing."},{"Start":"01:29.000 ","End":"01:30.560","Text":"Let\u0027s call it triangle."},{"Start":"01:30.560 ","End":"01:33.215","Text":"In this case, this is squared, this is the triangle."},{"Start":"01:33.215 ","End":"01:36.380","Text":"I noticed the same a on both sides."},{"Start":"01:36.380 ","End":"01:38.285","Text":"If you have this situation,"},{"Start":"01:38.285 ","End":"01:44.225","Text":"then we conclude that square equals triangle."},{"Start":"01:44.225 ","End":"01:46.999","Text":"That\u0027s the general rule we\u0027re going to be using,"},{"Start":"01:46.999 ","End":"01:48.440","Text":"at least in the beginning,"},{"Start":"01:48.440 ","End":"01:51.800","Text":"to solve a certain class of exponential equations."},{"Start":"01:51.800 ","End":"01:53.840","Text":"It\u0027s won\u0027t solve all exponential equations,"},{"Start":"01:53.840 ","End":"01:55.625","Text":"but it will solve many of them."},{"Start":"01:55.625 ","End":"01:58.340","Text":"Let\u0027s continue with this particular example."},{"Start":"01:58.340 ","End":"02:00.815","Text":"Hope you remember linear equations."},{"Start":"02:00.815 ","End":"02:02.300","Text":"We bring the x\u0027s to the left,"},{"Start":"02:02.300 ","End":"02:03.500","Text":"numbers to the right,"},{"Start":"02:03.500 ","End":"02:11.135","Text":"so we would get 5x minus 3x and on the other side we would get 7 plus 3."},{"Start":"02:11.135 ","End":"02:14.270","Text":"Then we would collect like terms and say,"},{"Start":"02:14.270 ","End":"02:17.690","Text":"okay, 5x minus 3x is 2x,"},{"Start":"02:17.690 ","End":"02:19.580","Text":"7 plus 3 is 10."},{"Start":"02:19.580 ","End":"02:23.420","Text":"Then we would divide by 2 and say x=5 is"},{"Start":"02:23.420 ","End":"02:28.010","Text":"a solution to this exponential equation and we would be done."},{"Start":"02:28.010 ","End":"02:29.630","Text":"I\u0027d also like to remind you,"},{"Start":"02:29.630 ","End":"02:33.725","Text":"I won\u0027t do it this time that when you get to the solution,"},{"Start":"02:33.725 ","End":"02:40.174","Text":"it\u0027s worthwhile substituting in the original equation and seeing that it actually works,"},{"Start":"02:40.174 ","End":"02:43.280","Text":"I\u0027ll just briefly go through that with you."},{"Start":"02:43.280 ","End":"02:45.950","Text":"Mentally, if x is 5,"},{"Start":"02:45.950 ","End":"02:49.050","Text":"5x minus 3 is 25,"},{"Start":"02:49.050 ","End":"02:51.240","Text":"minus 3 is 22."},{"Start":"02:51.240 ","End":"02:53.240","Text":"This says 3 to the 22."},{"Start":"02:53.240 ","End":"02:54.620","Text":"It\u0027s a huge number."},{"Start":"02:54.620 ","End":"02:57.725","Text":"Even a calculator won\u0027t show you to the end."},{"Start":"02:57.725 ","End":"03:02.640","Text":"Here we have 3 times 5 plus 7 is also 15,"},{"Start":"03:02.640 ","End":"03:04.244","Text":"plus 7 is 22."},{"Start":"03:04.244 ","End":"03:07.440","Text":"Here also 3 to the power of 22."},{"Start":"03:07.440 ","End":"03:09.230","Text":"It is indeed equal."},{"Start":"03:09.230 ","End":"03:11.810","Text":"Of course, the 22 is just the solution."},{"Start":"03:11.810 ","End":"03:15.020","Text":"I mean, it\u0027s what we get on the left-hand side here when x is 5 and"},{"Start":"03:15.020 ","End":"03:19.400","Text":"here also when x is 5, we get 22=22."},{"Start":"03:19.400 ","End":"03:21.730","Text":"Let\u0027s go for another example."},{"Start":"03:21.730 ","End":"03:26.990","Text":"Two to the power of 2x=1 over"},{"Start":"03:26.990 ","End":"03:33.305","Text":"32 times 1/8 to the power of x."},{"Start":"03:33.305 ","End":"03:35.540","Text":"Once again, it\u0027s an exponential equation."},{"Start":"03:35.540 ","End":"03:37.160","Text":"I see x in the exponent,"},{"Start":"03:37.160 ","End":"03:39.530","Text":"but this time it\u0027s not so simple."},{"Start":"03:39.530 ","End":"03:43.370","Text":"It\u0027s not like here where we had the same base and an exponent."},{"Start":"03:43.370 ","End":"03:46.790","Text":"We have to work a bit more to get to this point."},{"Start":"03:46.790 ","End":"03:50.375","Text":"However, there are no pluses and minuses here."},{"Start":"03:50.375 ","End":"03:54.560","Text":"It\u0027s just an exponent and then maybe multiplied or divided by something."},{"Start":"03:54.560 ","End":"03:56.285","Text":"That\u0027s going to be the typical case."},{"Start":"03:56.285 ","End":"03:59.930","Text":"Each time it\u0027ll get a little bit more difficult and you have to do a bit more work."},{"Start":"03:59.930 ","End":"04:03.650","Text":"But ultimately we will end up this paradigm, this template,"},{"Start":"04:03.650 ","End":"04:08.900","Text":"this formula where we have the same base and 2 exponents that we compare."},{"Start":"04:08.900 ","End":"04:11.270","Text":"Now here, I\u0027m looking first of all at the numbers."},{"Start":"04:11.270 ","End":"04:15.360","Text":"I see there\u0027s a 2, there is a 32 and there\u0027s an 8 and earlier,"},{"Start":"04:15.360 ","End":"04:17.870","Text":"I\u0027ve seen these numbers like 8 and 32 before."},{"Start":"04:17.870 ","End":"04:19.775","Text":"I know they are powers of 2,"},{"Start":"04:19.775 ","End":"04:21.920","Text":"so one of the things we would do would be to"},{"Start":"04:21.920 ","End":"04:24.550","Text":"try and get everything in terms of powers of 2."},{"Start":"04:24.550 ","End":"04:28.220","Text":"The left-hand side is already in a fairly straightforward form,"},{"Start":"04:28.220 ","End":"04:30.590","Text":"like a to the power of something where a is 2."},{"Start":"04:30.590 ","End":"04:35.480","Text":"Second equation will be 2 to the power of"},{"Start":"04:35.480 ","End":"04:43.804","Text":"2x=32 times 1/8 to the power of x."},{"Start":"04:43.804 ","End":"04:45.905","Text":"A little bit more complicated,"},{"Start":"04:45.905 ","End":"04:48.050","Text":"not straightforward like this,"},{"Start":"04:48.050 ","End":"04:51.739","Text":"not already in the form of a to the something equals a to the something."},{"Start":"04:51.739 ","End":"04:55.040","Text":"We have to just work a little bit to get it to that form."},{"Start":"04:55.040 ","End":"04:56.360","Text":"The left-hand side is okay,"},{"Start":"04:56.360 ","End":"04:57.980","Text":"2 to the power of something."},{"Start":"04:57.980 ","End":"05:02.240","Text":"Notice that this 2 indirectly appears on the right"},{"Start":"05:02.240 ","End":"05:06.590","Text":"because we know that 32 is 2 to the fifth,"},{"Start":"05:06.590 ","End":"05:10.055","Text":"and I know that 8 is 2 cubed."},{"Start":"05:10.055 ","End":"05:12.890","Text":"You\u0027ve seen these numbers before already."},{"Start":"05:12.890 ","End":"05:14.300","Text":"We should be familiar with this."},{"Start":"05:14.300 ","End":"05:18.665","Text":"I\u0027m going to just work a bit using rules of exponents to get it to the form,"},{"Start":"05:18.665 ","End":"05:20.840","Text":"also 2 to the power of something."},{"Start":"05:20.840 ","End":"05:24.545","Text":"The first step, I\u0027ll just write 32 is 2 to the fifth,"},{"Start":"05:24.545 ","End":"05:26.345","Text":"just as I mentioned here."},{"Start":"05:26.345 ","End":"05:28.730","Text":"Now, what do I do with 1/8?"},{"Start":"05:28.730 ","End":"05:38.420","Text":"Now 8 is 2 to the 3 and there is rule that says that 1 over something is the reciprocal,"},{"Start":"05:38.420 ","End":"05:40.700","Text":"so you make the exponent negative."},{"Start":"05:40.700 ","End":"05:42.560","Text":"Let me just write what I mean."},{"Start":"05:42.560 ","End":"05:47.555","Text":"This is 2 to the minus 3 to the power of x. I\u0027ll just quote the rule."},{"Start":"05:47.555 ","End":"05:53.030","Text":"The rule I looked it up was number 7 and it said that a to"},{"Start":"05:53.030 ","End":"05:59.115","Text":"the minus n is 1 over a to the n. For example,"},{"Start":"05:59.115 ","End":"06:04.285","Text":"here, 2 to the minus 3 is 1 over 2 to the 3."},{"Start":"06:04.285 ","End":"06:05.990","Text":"This is what enabled me,"},{"Start":"06:05.990 ","End":"06:08.135","Text":"if I read it from right to left,"},{"Start":"06:08.135 ","End":"06:10.000","Text":"to write 1 over 8,"},{"Start":"06:10.000 ","End":"06:14.475","Text":"which is 1 over 2 to the 3 as 2 to the minus 3."},{"Start":"06:14.475 ","End":"06:18.890","Text":"Another rule we\u0027re going to use is rule number 3,"},{"Start":"06:18.890 ","End":"06:27.260","Text":"which said that a to the power of m to the power of n. That\u0027s rule number 3,"},{"Start":"06:27.260 ","End":"06:31.310","Text":"is a to the power of m times n"},{"Start":"06:31.310 ","End":"06:35.620","Text":"and what it\u0027s going to mean here is just on the second part,"},{"Start":"06:35.620 ","End":"06:38.475","Text":"2 to the minus 3 to the power of x."},{"Start":"06:38.475 ","End":"06:43.020","Text":"This part will be 2 to the power of minus 3 times x,"},{"Start":"06:43.020 ","End":"06:44.590","Text":"which is minus 3x."},{"Start":"06:44.590 ","End":"06:47.390","Text":"The first part I\u0027m just copying as is."},{"Start":"06:47.390 ","End":"06:49.400","Text":"Now I get to this point."},{"Start":"06:49.400 ","End":"06:53.465","Text":"Now I apply yet another rule, the product rule,"},{"Start":"06:53.465 ","End":"06:56.105","Text":"which says that when we have the same base"},{"Start":"06:56.105 ","End":"06:58.850","Text":"and we multiply and these 2 different exponents,"},{"Start":"06:58.850 ","End":"07:01.115","Text":"we add those exponents."},{"Start":"07:01.115 ","End":"07:05.270","Text":"This was actually our very first rule, number 1,"},{"Start":"07:05.270 ","End":"07:13.205","Text":"that a to the power of m plus n is a to the m times a to the power of n. In our case,"},{"Start":"07:13.205 ","End":"07:20.680","Text":"it means that we add this is my m and this is my n. We get 2 to the power of 5 minus 3x."},{"Start":"07:20.680 ","End":"07:23.795","Text":"I mean, I\u0027m adding but it\u0027s a negative number, it\u0027s minus."},{"Start":"07:23.795 ","End":"07:30.410","Text":"Okay, now let me just bring the left-hand side down and we have 2 to the 2x equals this."},{"Start":"07:30.410 ","End":"07:32.780","Text":"We\u0027re now at the point which we wanted,"},{"Start":"07:32.780 ","End":"07:34.835","Text":"which is this. I\u0027ll emphasize it."},{"Start":"07:34.835 ","End":"07:38.285","Text":"We have the same base 2 on both sides,"},{"Start":"07:38.285 ","End":"07:40.685","Text":"just like here we have a and a,"},{"Start":"07:40.685 ","End":"07:42.890","Text":"this is my square, this is a triangle."},{"Start":"07:42.890 ","End":"07:49.330","Text":"We get 2x=5-3x."},{"Start":"07:49.330 ","End":"07:51.199","Text":"You just throw out the bases."},{"Start":"07:51.199 ","End":"07:54.395","Text":"But what we\u0027re really doing is comparing the exponents."},{"Start":"07:54.395 ","End":"07:56.119","Text":"Okay, from here I\u0027ll continue."},{"Start":"07:56.119 ","End":"07:58.040","Text":"It\u0027s just straightforward."},{"Start":"07:58.040 ","End":"07:59.660","Text":"Bring the x\u0027s to the left,"},{"Start":"07:59.660 ","End":"08:00.710","Text":"numbers to the right."},{"Start":"08:00.710 ","End":"08:04.770","Text":"So what I have on the left is 2x plus 3x on the right,"},{"Start":"08:04.770 ","End":"08:08.730","Text":"5 remains, 2x and 3x together is 5x."},{"Start":"08:08.730 ","End":"08:13.140","Text":"If I divide both sides by 5, I\u0027ve got x=1."},{"Start":"08:13.140 ","End":"08:15.025","Text":"This is the answer to this one,"},{"Start":"08:15.025 ","End":"08:21.650","Text":"and we\u0027ll just do a quick check mentally to see that this indeed works."},{"Start":"08:21.650 ","End":"08:24.440","Text":"Let\u0027s see, on the left we have 2 to the power of 2x,"},{"Start":"08:24.440 ","End":"08:26.880","Text":"which is 2 to the power of 2,"},{"Start":"08:26.880 ","End":"08:28.365","Text":"because 2 times 1 is 2."},{"Start":"08:28.365 ","End":"08:31.045","Text":"Two to the power of 2 is 4."},{"Start":"08:31.045 ","End":"08:34.475","Text":"Let me just make a note of that somewhere it\u0027s 4,"},{"Start":"08:34.475 ","End":"08:37.249","Text":"if x is 1 and on the right-hand side,"},{"Start":"08:37.249 ","End":"08:38.600","Text":"we have, let\u0027s see,"},{"Start":"08:38.600 ","End":"08:46.860","Text":"1/8 to the power of 1 is just an 1\\8 and 32 times an 1\\8 is 32 divided by 8,"},{"Start":"08:46.860 ","End":"08:49.565","Text":"which is also 4, so yes,"},{"Start":"08:49.565 ","End":"08:54.870","Text":"this is verified, just like we mentally verified this 1."},{"Start":"08:54.870 ","End":"08:57.050","Text":"That\u0027s something you should get into the habit of doing,"},{"Start":"08:57.050 ","End":"08:59.435","Text":"if not always, at least once in a while."},{"Start":"08:59.435 ","End":"09:04.460","Text":"We will settle for these 2 examples to illustrate this principle."},{"Start":"09:04.460 ","End":"09:08.840","Text":"The solved exercises that we have as part of the course,"},{"Start":"09:08.840 ","End":"09:12.335","Text":"we\u0027ll continue and get a bit more complicated than this."},{"Start":"09:12.335 ","End":"09:14.570","Text":"In the next clip on exponential equations,"},{"Start":"09:14.570 ","End":"09:17.950","Text":"we\u0027ll talk about a different principle other than this one."},{"Start":"09:17.950 ","End":"09:20.600","Text":"Here we\u0027re done for now."}],"ID":8115},{"Watched":false,"Name":"Exercise 1","Duration":"11m 18s","ChapterTopicVideoID":8025,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8025.jpeg","UploadDate":"2020-09-30T14:21:29.4900000","DurationForVideoObject":"PT11M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.790","Text":"In this exercise, we have some exponential equations to solve an exponent."},{"Start":"00:05.790 ","End":"00:08.805","Text":"For example, in the first 1 we have something to the power of something."},{"Start":"00:08.805 ","End":"00:11.160","Text":"Remember this part is called the base,"},{"Start":"00:11.160 ","End":"00:16.170","Text":"and this part at the top is called the exponent base to the power of exponent."},{"Start":"00:16.170 ","End":"00:21.255","Text":"The general idea in all of these exercises is the same."},{"Start":"00:21.255 ","End":"00:23.295","Text":"We want to bring it to the form,"},{"Start":"00:23.295 ","End":"00:27.060","Text":"some base to the power of some exponent,"},{"Start":"00:27.060 ","End":"00:32.670","Text":"let\u0027s called an exponent 1 is equal to the same base,"},{"Start":"00:32.670 ","End":"00:38.100","Text":"that\u0027s important, same base to the power of another exponent, exponent 2."},{"Start":"00:38.100 ","End":"00:42.070","Text":"If I get to this situation and let me frame it,"},{"Start":"00:42.070 ","End":"00:46.625","Text":"we conclude that the 2 exponents are equal."},{"Start":"00:46.625 ","End":"00:52.775","Text":"That exponent 1 is equal to exponent 2."},{"Start":"00:52.775 ","End":"00:58.760","Text":"It\u0027s as if we had canceled the bases and just compared the exponents."},{"Start":"00:58.760 ","End":"01:01.445","Text":"Typically, this base will be a number like,"},{"Start":"01:01.445 ","End":"01:07.445","Text":"2 or 3 and the exponent will usually be an expression involving x or some other letter."},{"Start":"01:07.445 ","End":"01:10.190","Text":"Let\u0027s see how it works out in the examples."},{"Start":"01:10.190 ","End":"01:12.140","Text":"In the first example on the left,"},{"Start":"01:12.140 ","End":"01:19.370","Text":"we already have an exponent power like this is 2 is the base and 3x is the exponent."},{"Start":"01:19.370 ","End":"01:21.860","Text":"We would like to get the same thing on the right."},{"Start":"01:21.860 ","End":"01:26.195","Text":"Let\u0027s see, can we write this as 2 to the power of something here?"},{"Start":"01:26.195 ","End":"01:31.130","Text":"I do a little side exercise and start multiplying 2 times 2."},{"Start":"01:31.130 ","End":"01:32.750","Text":"Let\u0027s see 2, we get 4,"},{"Start":"01:32.750 ","End":"01:37.025","Text":"we\u0027ve got 8,16,"},{"Start":"01:37.025 ","End":"01:39.415","Text":"32, 64, good."},{"Start":"01:39.415 ","End":"01:41.190","Text":"Then we see the 6 of them,"},{"Start":"01:41.190 ","End":"01:45.685","Text":"so we write that 64 is 2^6."},{"Start":"01:45.685 ","End":"01:47.120","Text":"Some of you will remember this,"},{"Start":"01:47.120 ","End":"01:49.475","Text":"some of you\u0027ll get used to the powers of 2,"},{"Start":"01:49.475 ","End":"01:52.680","Text":"but if not you just keep multiplying by 2 by"},{"Start":"01:52.680 ","End":"01:57.440","Text":"2 until you get 64 and then count how many \u00272s you have."},{"Start":"01:57.440 ","End":"02:00.980","Text":"Now, using this paradigm, this model,"},{"Start":"02:00.980 ","End":"02:04.459","Text":"if I have the same base and then 2 different exponents,"},{"Start":"02:04.459 ","End":"02:06.560","Text":"it look different anyway,"},{"Start":"02:06.560 ","End":"02:08.705","Text":"then we equate the exponents."},{"Start":"02:08.705 ","End":"02:12.355","Text":"We say now that 3x is equal to 6,"},{"Start":"02:12.355 ","End":"02:13.880","Text":"and from here you know what to do."},{"Start":"02:13.880 ","End":"02:20.660","Text":"We divide both sides by 3 and we get x equals 2 That\u0027s the answer to the first 1,"},{"Start":"02:20.660 ","End":"02:24.150","Text":"so onto the next 1."},{"Start":"02:24.230 ","End":"02:28.670","Text":"This is similar to the above situation here we already have"},{"Start":"02:28.670 ","End":"02:33.525","Text":"a base 5 to the power of some exponent 3x minus 6,"},{"Start":"02:33.525 ","End":"02:35.960","Text":"we would like the same thing on the right."},{"Start":"02:35.960 ","End":"02:39.890","Text":"We try to see if this can be 5 to the power of something."},{"Start":"02:39.890 ","End":"02:41.449","Text":"If you don\u0027t see it straightaway,"},{"Start":"02:41.449 ","End":"02:43.360","Text":"you just start multiplying 5."},{"Start":"02:43.360 ","End":"02:45.240","Text":"We\u0027ve got just 1 of them it\u0027s 5,"},{"Start":"02:45.240 ","End":"02:50.245","Text":"another 5 it\u0027s 25 already times 5 again, 125."},{"Start":"02:50.245 ","End":"02:51.770","Text":"Yes, that\u0027s what we wanted."},{"Start":"02:51.770 ","End":"02:53.390","Text":"Count how many 5s we have,"},{"Start":"02:53.390 ","End":"02:54.865","Text":"1, 2, 3."},{"Start":"02:54.865 ","End":"02:57.825","Text":"So 125 is 5^3."},{"Start":"02:57.825 ","End":"03:00.980","Text":"Then using that same principle that if we have the same base,"},{"Start":"03:00.980 ","End":"03:03.095","Text":"it\u0027s as if we can throw the base out."},{"Start":"03:03.095 ","End":"03:10.230","Text":"Throw out the 5s and we\u0027ve got 3x minus 6 equals 3 linear equation."},{"Start":"03:10.230 ","End":"03:12.690","Text":"Bring the 6 to the other side."},{"Start":"03:12.690 ","End":"03:17.385","Text":"3x equals 3 plus 6 is 9,"},{"Start":"03:17.385 ","End":"03:20.400","Text":"and x equals 3."},{"Start":"03:20.400 ","End":"03:22.950","Text":"That\u0027s part b solved."},{"Start":"03:22.950 ","End":"03:28.964","Text":"Moving on, let\u0027s move on to c and d. Let\u0027s do part d first."},{"Start":"03:28.964 ","End":"03:33.164","Text":"The reason I prefer part d is that in this case,"},{"Start":"03:33.164 ","End":"03:36.600","Text":"49 is 7^2,"},{"Start":"03:36.600 ","End":"03:38.780","Text":"just 1 of those things you know that 7 times 7 is"},{"Start":"03:38.780 ","End":"03:42.835","Text":"49 and 1 of them is a power of the other,"},{"Start":"03:42.835 ","End":"03:44.810","Text":"it turns out to be easier."},{"Start":"03:44.810 ","End":"03:46.910","Text":"There\u0027s a slight extra complication in c,"},{"Start":"03:46.910 ","End":"03:49.175","Text":"which is why I want to do d first."},{"Start":"03:49.175 ","End":"03:52.025","Text":"If I write 49 as 7^2,"},{"Start":"03:52.025 ","End":"03:55.460","Text":"7 times 7 is 7^2."},{"Start":"03:55.460 ","End":"04:05.595","Text":"You can write it as (7^2) x minus 2 equals 7^3x minus 4."},{"Start":"04:05.595 ","End":"04:10.260","Text":"Now, recall the formulas for exponents."},{"Start":"04:10.260 ","End":"04:16.535","Text":"What I have prepared page and here is the formula sheet for exponents."},{"Start":"04:16.535 ","End":"04:21.610","Text":"What I have here is (7^2)^x minus 2."},{"Start":"04:21.610 ","End":"04:25.340","Text":"This looks very much like this."},{"Start":"04:25.340 ","End":"04:29.284","Text":"Something to the power of something and then to the power of something."},{"Start":"04:29.284 ","End":"04:32.210","Text":"This is a formula I\u0027m going to be using,"},{"Start":"04:32.210 ","End":"04:36.830","Text":"but I\u0027m going to be reading it from right to left because this is the form I have."},{"Start":"04:36.830 ","End":"04:41.990","Text":"Going back here 2 is my m and x minus 2 is my n. I"},{"Start":"04:41.990 ","End":"04:48.475","Text":"get 7^2 (x minus 2)."},{"Start":"04:48.475 ","End":"04:53.150","Text":"I need to put the x minus 2 in brackets because it\u0027s 2 times the whole thing is"},{"Start":"04:53.150 ","End":"04:58.170","Text":"equal to 7^3x minus 4."},{"Start":"04:58.170 ","End":"05:02.345","Text":"Now I can use the principal I had above,"},{"Start":"05:02.345 ","End":"05:04.250","Text":"where if I have the same base,"},{"Start":"05:04.250 ","End":"05:05.870","Text":"7 is the same base,"},{"Start":"05:05.870 ","End":"05:08.555","Text":"I can just equate the exponents."},{"Start":"05:08.555 ","End":"05:16.195","Text":"I get that twice x minus 2 equals 3x minus 4."},{"Start":"05:16.195 ","End":"05:19.130","Text":"At this point it\u0027s no longer an exponential equation."},{"Start":"05:19.130 ","End":"05:22.205","Text":"It\u0027s applying linear equation in 1 variable."},{"Start":"05:22.205 ","End":"05:25.595","Text":"Let me continue with the solution over here."},{"Start":"05:25.595 ","End":"05:34.855","Text":"Opening the brackets, it\u0027s 2x minus 4 equals 3x minus 4,"},{"Start":"05:34.855 ","End":"05:38.015","Text":"bringing the xs to the left and the numbers to the right,"},{"Start":"05:38.015 ","End":"05:45.705","Text":"I get 2x minus 3x equals minus 4 plus 4."},{"Start":"05:45.705 ","End":"05:51.360","Text":"Let\u0027s see, 2x minus 3x is minus 1x or just minus x,"},{"Start":"05:51.360 ","End":"05:55.350","Text":"minus 4 plus 4 is 0, minus x is 0,"},{"Start":"05:55.350 ","End":"05:57.830","Text":"then obviously x is 0 also,"},{"Start":"05:57.830 ","End":"06:03.125","Text":"and that\u0027s the solution for part d. Now,"},{"Start":"06:03.125 ","End":"06:05.900","Text":"let\u0027s get back to c,"},{"Start":"06:05.900 ","End":"06:09.020","Text":"thing here is that I couldn\u0027t say that 9 is"},{"Start":"06:09.020 ","End":"06:12.275","Text":"27 to the something or 27 is 9 to the something."},{"Start":"06:12.275 ","End":"06:13.910","Text":"But if we can\u0027t do that,"},{"Start":"06:13.910 ","End":"06:17.590","Text":"we can find 1/3 thing that both the powers of it."},{"Start":"06:17.590 ","End":"06:22.095","Text":"The obvious candidate to try is 3."},{"Start":"06:22.095 ","End":"06:25.620","Text":"I want to try and write 9 and 27,"},{"Start":"06:25.620 ","End":"06:29.860","Text":"as I said both to the power of the same thing and we\u0027re proposing 3."},{"Start":"06:29.860 ","End":"06:35.805","Text":"3 to the something here and 3 to the something here."},{"Start":"06:35.805 ","End":"06:39.015","Text":"Well, we know that 9 is 3 times 3."},{"Start":"06:39.015 ","End":"06:42.510","Text":"This question mark is 2 and as for this"},{"Start":"06:42.510 ","End":"06:47.235","Text":"27 is 3 times 3 times 3 so this question mark is 3."},{"Start":"06:47.235 ","End":"06:52.010","Text":"Now back here, I\u0027m going to put everything in terms of base 3."},{"Start":"06:52.010 ","End":"07:00.185","Text":"First thing I do is just write 9 as 3^2 like I saw here and then the x minus 1 stays."},{"Start":"07:00.185 ","End":"07:07.605","Text":"27 is 3^3 and the x plus 2, just days."},{"Start":"07:07.605 ","End":"07:11.990","Text":"Now I\u0027m going to use again this formula twice,"},{"Start":"07:11.990 ","End":"07:17.455","Text":"once on the left side and once for the right-hand side but again from right to left."},{"Start":"07:17.455 ","End":"07:22.580","Text":"What I get here is 3 to the power of this times this."},{"Start":"07:22.580 ","End":"07:31.095","Text":"In other words, twice x minus 1 equals 3^3 times x plus 2."},{"Start":"07:31.095 ","End":"07:34.010","Text":"Now we\u0027ll use that most basic principle for"},{"Start":"07:34.010 ","End":"07:37.280","Text":"all these exercises is if we have the same base,"},{"Start":"07:37.280 ","End":"07:39.545","Text":"we compare the exponents."},{"Start":"07:39.545 ","End":"07:48.430","Text":"We get twice x minus 1 equals 3 times x plus 2."},{"Start":"07:48.430 ","End":"07:52.580","Text":"Here once again we have a linear equation."},{"Start":"07:52.580 ","End":"07:55.115","Text":"Let me just continue over here."},{"Start":"07:55.115 ","End":"07:59.455","Text":"If I continue, I get expanding"},{"Start":"07:59.455 ","End":"08:07.770","Text":"2x minus 2 equals 3x plus 6,"},{"Start":"08:07.770 ","End":"08:10.320","Text":"x is to the left, numbers to the right,"},{"Start":"08:10.320 ","End":"08:15.945","Text":"2x minus 3x is minus x and 6 plus"},{"Start":"08:15.945 ","End":"08:22.735","Text":"2 is 8 and so multiplying or dividing by minus 1,"},{"Start":"08:22.735 ","End":"08:31.295","Text":"we get that x is minus 8 and that\u0027s the answer to part c. We\u0027re basically done,"},{"Start":"08:31.295 ","End":"08:33.650","Text":"but from time to time,"},{"Start":"08:33.650 ","End":"08:39.350","Text":"or even every time you should check your answer by substituting."},{"Start":"08:39.350 ","End":"08:43.385","Text":"Let\u0027s say we want to check in part b."},{"Start":"08:43.385 ","End":"08:49.760","Text":"Let\u0027s put x equals 3 in the original equation and see what we get."},{"Start":"08:49.760 ","End":"08:52.175","Text":"If x equals 3,"},{"Start":"08:52.175 ","End":"08:58.640","Text":"then 5^3x minus 6."},{"Start":"08:58.640 ","End":"09:02.990","Text":"I can just work on the left-hand side and see if I reach the right-hand side."},{"Start":"09:02.990 ","End":"09:05.105","Text":"That\u0027s 1 way of checking equality."},{"Start":"09:05.105 ","End":"09:12.230","Text":"So let\u0027s see, this is equal to 5^3 times 3 minus 6,"},{"Start":"09:12.230 ","End":"09:16.100","Text":"which is equal to 3 times 3 is 9."},{"Start":"09:16.100 ","End":"09:20.600","Text":"9 minus 6 is 3, is 5^3,"},{"Start":"09:20.600 ","End":"09:24.130","Text":"which is 5 times 5 times 5,"},{"Start":"09:24.130 ","End":"09:26.505","Text":"which is 125,"},{"Start":"09:26.505 ","End":"09:29.100","Text":"which is the right-hand side."},{"Start":"09:29.100 ","End":"09:32.270","Text":"So x equals 3 is being checked."},{"Start":"09:32.270 ","End":"09:34.130","Text":"Let\u0027s just try 1 more."},{"Start":"09:34.130 ","End":"09:37.385","Text":"I\u0027ll try the last 1 and see if x equals 0 works."},{"Start":"09:37.385 ","End":"09:39.499","Text":"If I substitute in the original,"},{"Start":"09:39.499 ","End":"09:46.400","Text":"I get 49^0 minus 2."},{"Start":"09:46.400 ","End":"09:50.030","Text":"This time I\u0027ll write the other side also but I\u0027ll put"},{"Start":"09:50.030 ","End":"09:53.705","Text":"a question mark because we\u0027re checking, is it equal?"},{"Start":"09:53.705 ","End":"10:02.750","Text":"Let\u0027s verify if this is equal to 7^3 times 0 minus 4."},{"Start":"10:02.750 ","End":"10:08.960","Text":"Let\u0027s see what we get. Here we get 49 to the power minus"},{"Start":"10:08.960 ","End":"10:15.140","Text":"2 equals 7 to the power of minus 4."},{"Start":"10:15.140 ","End":"10:20.780","Text":"1 of the formulas that I could use is this 1 here."},{"Start":"10:20.780 ","End":"10:23.375","Text":"How to deal with a negative exponent."},{"Start":"10:23.375 ","End":"10:26.195","Text":"This set of have a negative exponent,"},{"Start":"10:26.195 ","End":"10:28.400","Text":"and I can make it a positive exponent,"},{"Start":"10:28.400 ","End":"10:30.670","Text":"but in the denominator."},{"Start":"10:30.670 ","End":"10:36.465","Text":"I get this is equal to 1 over using this formula,"},{"Start":"10:36.465 ","End":"10:39.245","Text":"49^2 and on the right-hand side,"},{"Start":"10:39.245 ","End":"10:44.660","Text":"equals with a question mark, 1/7^4."},{"Start":"10:44.660 ","End":"10:50.810","Text":"At this point, you could either use the calculator or you could multiply 49 times 49."},{"Start":"10:50.810 ","End":"10:57.930","Text":"Left-hand side comes out to 1/2,401,"},{"Start":"10:57.930 ","End":"11:02.015","Text":"and if you do 7^4 on the calculator or in your head,"},{"Start":"11:02.015 ","End":"11:08.310","Text":"or on paper, you also get 1/2,401."},{"Start":"11:08.310 ","End":"11:14.915","Text":"The equality really works and so we\u0027ve verified that this is a solution."},{"Start":"11:14.915 ","End":"11:18.810","Text":"We\u0027re done for this set of exercise."}],"ID":8118},{"Watched":false,"Name":"Exercise 2","Duration":"6m 31s","ChapterTopicVideoID":8026,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8026.jpeg","UploadDate":"2020-09-30T14:27:11.6670000","DurationForVideoObject":"PT6M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"Here we have a couple of exponential equations that we need to solve"},{"Start":"00:04.560 ","End":"00:11.130","Text":"and I even prepared the formula sheet for exponents in case we need it."},{"Start":"00:11.130 ","End":"00:13.410","Text":"Let\u0027s start with the first one."},{"Start":"00:13.410 ","End":"00:19.155","Text":"Remember, the idea is to get both sides as an exponent with the same base."},{"Start":"00:19.155 ","End":"00:22.080","Text":"Question is what would that same base be?"},{"Start":"00:22.080 ","End":"00:24.450","Text":"Well, 8 is a power of 2,"},{"Start":"00:24.450 ","End":"00:30.255","Text":"it\u0027s 2^3, so 2 will do to convert both sides to base 2."},{"Start":"00:30.255 ","End":"00:35.535","Text":"I\u0027ll just make a note that 8 is 2^3,"},{"Start":"00:35.535 ","End":"00:39.900","Text":"and that we\u0027re going to use base 2 for both sides."},{"Start":"00:39.900 ","End":"00:41.645","Text":"The left hand side,"},{"Start":"00:41.645 ","End":"00:45.230","Text":"I just substitute 8 is 2^3."},{"Start":"00:45.230 ","End":"00:53.340","Text":"We get 2^3 to the power of 4x-2 equals."},{"Start":"00:53.340 ","End":"00:56.065","Text":"Now what am I going to do with the 1/2?"},{"Start":"00:56.065 ","End":"00:58.190","Text":"Going to use this formula here,"},{"Start":"00:58.190 ","End":"01:01.475","Text":"but I\u0027m going to use this formula from right to left."},{"Start":"01:01.475 ","End":"01:04.215","Text":"If I take n=1 here,"},{"Start":"01:04.215 ","End":"01:08.565","Text":"then what we have is exactly 1/2 ^1."},{"Start":"01:08.565 ","End":"01:18.395","Text":"Using this formula, that\u0027s 2^minus 1 and then to the power of 3x minus 3 as here."},{"Start":"01:18.395 ","End":"01:20.225","Text":"I\u0027ll go over this last bit again."},{"Start":"01:20.225 ","End":"01:21.610","Text":"If n is 1,"},{"Start":"01:21.610 ","End":"01:27.175","Text":"I get 1/2 ^1 is 2^minus 1,"},{"Start":"01:27.175 ","End":"01:28.515","Text":"a is 2 here."},{"Start":"01:28.515 ","End":"01:32.240","Text":"Next step is to use another formula."},{"Start":"01:32.240 ","End":"01:36.485","Text":"We use this one a lot and we also use it from right to left."},{"Start":"01:36.485 ","End":"01:38.015","Text":"What do we have here?"},{"Start":"01:38.015 ","End":"01:44.520","Text":"We have 2^3 times 4x-2 for this one."},{"Start":"01:44.520 ","End":"01:47.310","Text":"I\u0027m going to use the formula again for this one,"},{"Start":"01:47.310 ","End":"01:53.280","Text":"and I get 2^ minus 1 times 3x-3."},{"Start":"01:53.280 ","End":"01:56.480","Text":"What this formula says basically if I read it from right to left,"},{"Start":"01:56.480 ","End":"01:59.090","Text":"if I take an exponent of an exponent,"},{"Start":"01:59.090 ","End":"02:01.790","Text":"I just multiply those two exponents and that\u0027s what I did here,"},{"Start":"02:01.790 ","End":"02:03.860","Text":"I multiplied this by this here,"},{"Start":"02:03.860 ","End":"02:06.190","Text":"and I multiplied this by this here."},{"Start":"02:06.190 ","End":"02:08.720","Text":"Now, we\u0027re exactly where we want to be because we have"},{"Start":"02:08.720 ","End":"02:11.420","Text":"two exponents with the same base 2."},{"Start":"02:11.420 ","End":"02:16.350","Text":"When we have the same base then we just compare the exponents,"},{"Start":"02:16.350 ","End":"02:25.585","Text":"so 3 times 4x-2 is equal to minus 1 3x minus 3."},{"Start":"02:25.585 ","End":"02:29.860","Text":"I hit the formula so I bring them back as needed and get some more room here."},{"Start":"02:29.860 ","End":"02:31.850","Text":"Let\u0027s continue over here."},{"Start":"02:31.850 ","End":"02:39.230","Text":"Expanding brackets we get 12x-3 times 2 is 6,"},{"Start":"02:39.230 ","End":"02:42.050","Text":"is equal to minus 1."},{"Start":"02:42.050 ","End":"02:45.590","Text":"I\u0027ve got minus 3x+3,"},{"Start":"02:45.590 ","End":"02:46.865","Text":"x is to the left,"},{"Start":"02:46.865 ","End":"02:48.380","Text":"numbers to the right,"},{"Start":"02:48.380 ","End":"02:54.465","Text":"12x+3x=3+6,"},{"Start":"02:54.465 ","End":"02:59.650","Text":"12 and 3 is 15x=9,"},{"Start":"02:59.650 ","End":"03:02.960","Text":"dividing by 15 or bringing the 15 down on"},{"Start":"03:02.960 ","End":"03:07.320","Text":"the other side to the denominator we get x is 9/15."},{"Start":"03:07.640 ","End":"03:10.775","Text":"It\u0027s customary to reduce where possible."},{"Start":"03:10.775 ","End":"03:13.145","Text":"Since both are divisible by 3,"},{"Start":"03:13.145 ","End":"03:16.940","Text":"I\u0027ll divide this by 3 and this by 3,"},{"Start":"03:16.940 ","End":"03:20.745","Text":"so the answer is 3/5."},{"Start":"03:20.745 ","End":"03:22.095","Text":"This is our answer,"},{"Start":"03:22.095 ","End":"03:24.320","Text":"and onto the next one,"},{"Start":"03:24.320 ","End":"03:26.825","Text":"get some more room here."},{"Start":"03:26.825 ","End":"03:30.454","Text":"Here we have a 10 on the left,"},{"Start":"03:30.454 ","End":"03:34.070","Text":"and here we have some decimal on the right as the base."},{"Start":"03:34.070 ","End":"03:36.530","Text":"But a decimal is easily convertible to"},{"Start":"03:36.530 ","End":"03:39.890","Text":"a decimal fraction or a fraction involving powers of 10."},{"Start":"03:39.890 ","End":"03:41.570","Text":"Let\u0027s do that at the side."},{"Start":"03:41.570 ","End":"03:47.405","Text":"0.001, what we do is we count 1, 2,"},{"Start":"03:47.405 ","End":"03:50.999","Text":"3 places to the right of the decimal point,"},{"Start":"03:50.999 ","End":"03:53.565","Text":"3 places means it\u0027s a fraction,"},{"Start":"03:53.565 ","End":"03:55.845","Text":"1 with three zeros, 1,"},{"Start":"03:55.845 ","End":"03:58.840","Text":"2, 3, for each one of these I need a zero."},{"Start":"03:58.840 ","End":"04:01.100","Text":"Then this number just goes on the top."},{"Start":"04:01.100 ","End":"04:05.300","Text":"But I prefer to write this as 1/10 ^3."},{"Start":"04:05.300 ","End":"04:12.515","Text":"Going back here, I have on the left-hand side 10^6x-2."},{"Start":"04:12.515 ","End":"04:15.980","Text":"You know what? We should even continue further with this."},{"Start":"04:15.980 ","End":"04:19.430","Text":"I brought back what I hid before, the formula sheet."},{"Start":"04:19.430 ","End":"04:21.305","Text":"We\u0027re going to use this one."},{"Start":"04:21.305 ","End":"04:25.130","Text":"If we take a=10 and n=3,"},{"Start":"04:25.130 ","End":"04:30.665","Text":"we see that 1/10 ^3 is just 10^ minus 3."},{"Start":"04:30.665 ","End":"04:35.255","Text":"Instead of 0.001, I write 10^ minus 3,"},{"Start":"04:35.255 ","End":"04:42.840","Text":"all this to the power of 1.5x-2."},{"Start":"04:42.840 ","End":"04:47.270","Text":"Let\u0027s continue, and this time we\u0027re going to use this formula again,"},{"Start":"04:47.270 ","End":"04:48.430","Text":"from right to left."},{"Start":"04:48.430 ","End":"04:53.380","Text":"Left-hand side is okay, 10^6x minus 2."},{"Start":"04:53.380 ","End":"04:56.300","Text":"But here I\u0027m going to use the rule of the power of a power,"},{"Start":"04:56.300 ","End":"04:58.955","Text":"so you multiply the exponents."},{"Start":"04:58.955 ","End":"05:06.775","Text":"We\u0027ve got 10^ minus 3 times 1.5x-2."},{"Start":"05:06.775 ","End":"05:09.215","Text":"Now this is the point where we want to be."},{"Start":"05:09.215 ","End":"05:14.435","Text":"We have an equation of exponents with the same base 10."},{"Start":"05:14.435 ","End":"05:16.430","Text":"What we usually do in this case,"},{"Start":"05:16.430 ","End":"05:17.660","Text":"we compare the exponents,"},{"Start":"05:17.660 ","End":"05:26.575","Text":"so I have 6x-2 equals minus 3 times 1.5x-2."},{"Start":"05:26.575 ","End":"05:30.290","Text":"Then we multiply out on the right-hand side."},{"Start":"05:30.290 ","End":"05:33.170","Text":"The left-hand side is nothing to do."},{"Start":"05:33.170 ","End":"05:38.480","Text":"Here, minus 3 times 1.5 minus is 4.5x,"},{"Start":"05:38.480 ","End":"05:41.225","Text":"minus 3 times minus 2 is plus 6."},{"Start":"05:41.225 ","End":"05:43.029","Text":"Bring the xs to the left,"},{"Start":"05:43.029 ","End":"05:44.615","Text":"numbers to the right,"},{"Start":"05:44.615 ","End":"05:50.630","Text":"6x-4.5x equals, sorry, plus,"},{"Start":"05:50.630 ","End":"05:53.810","Text":"it\u0027s switched sides so it becomes a plus, of course."},{"Start":"05:53.810 ","End":"05:58.445","Text":"Here the 2 also becomes a plus, 6+2."},{"Start":"05:58.445 ","End":"06:05.495","Text":"What I have is 10.5x=8,"},{"Start":"06:05.495 ","End":"06:10.465","Text":"this is 10.5, 10.5x =8."},{"Start":"06:10.465 ","End":"06:15.110","Text":"Continuing over here, double both sides and we\u0027ll get rid of the fraction,"},{"Start":"06:15.110 ","End":"06:16.505","Text":"that seems to be the simplest."},{"Start":"06:16.505 ","End":"06:22.070","Text":"So 21x is equal to 16, and then x,"},{"Start":"06:22.070 ","End":"06:24.980","Text":"if I divide both sides by 21,"},{"Start":"06:24.980 ","End":"06:32.280","Text":"is 16/21, and this is the answer."}],"ID":8119},{"Watched":false,"Name":"Exercise 3","Duration":"11m 46s","ChapterTopicVideoID":8023,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8023.jpeg","UploadDate":"2020-09-30T14:34:11.5170000","DurationForVideoObject":"PT11M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"Here we have another couple of equations to solve."},{"Start":"00:03.210 ","End":"00:05.295","Text":"They both have exponents in them."},{"Start":"00:05.295 ","End":"00:07.455","Text":"Let\u0027s start with the first one."},{"Start":"00:07.455 ","End":"00:11.160","Text":"As usual, I have the formula sheet ready,"},{"Start":"00:11.160 ","End":"00:15.090","Text":"I even have the highlighting from the previous clips."},{"Start":"00:15.090 ","End":"00:17.250","Text":"We have two exponents,"},{"Start":"00:17.250 ","End":"00:19.320","Text":"but they have different bases."},{"Start":"00:19.320 ","End":"00:22.470","Text":"We want to try and get it to be the same base."},{"Start":"00:22.470 ","End":"00:25.960","Text":"Let\u0027s see if we can do some numeric manipulation."},{"Start":"00:25.960 ","End":"00:28.660","Text":"If I look at 25 over 9,"},{"Start":"00:28.660 ","End":"00:34.490","Text":"I see immediately that this is 5^2 over 3^2."},{"Start":"00:34.490 ","End":"00:36.170","Text":"You really should memorize some of"},{"Start":"00:36.170 ","End":"00:41.540","Text":"the smaller exponent squares and cubes and things like that,"},{"Start":"00:41.540 ","End":"00:43.025","Text":"it comes in very handy."},{"Start":"00:43.025 ","End":"00:44.240","Text":"The other one,"},{"Start":"00:44.240 ","End":"00:47.900","Text":"not as elementary but also fairly familiar,"},{"Start":"00:47.900 ","End":"00:50.780","Text":"you\u0027ve seen 125, you\u0027ve seen 27."},{"Start":"00:50.780 ","End":"00:57.650","Text":"This is 3 times 3 times 3 or 3^3 and this is 5 times 5 times 5 or 5^3."},{"Start":"00:57.650 ","End":"01:00.890","Text":"This time I\u0027d like to use another formulas here."},{"Start":"01:00.890 ","End":"01:03.250","Text":"I\u0027m referring to this one,"},{"Start":"01:03.250 ","End":"01:08.255","Text":"and for some reason we seem to be using them all backwards from right to left."},{"Start":"01:08.255 ","End":"01:14.905","Text":"In this case, I can write this as 5 over 3^2,"},{"Start":"01:14.905 ","End":"01:19.790","Text":"and here I can write 3 over 5^3."},{"Start":"01:19.790 ","End":"01:22.520","Text":"Now, in our quest to find a common base,"},{"Start":"01:22.520 ","End":"01:29.020","Text":"we\u0027ve come close because we see that here we have a 5 over 3 and here we have a 3 over 5."},{"Start":"01:29.020 ","End":"01:31.615","Text":"What\u0027s the relationship between these 2?"},{"Start":"01:31.615 ","End":"01:34.055","Text":"You probably know that these are reciprocals."},{"Start":"01:34.055 ","End":"01:36.380","Text":"Each one is one over the other,"},{"Start":"01:36.380 ","End":"01:40.351","Text":"because whenever we take one over a fraction, we invert it."},{"Start":"01:40.351 ","End":"01:42.740","Text":"Let\u0027s decide which one is the right way up."},{"Start":"01:42.740 ","End":"01:46.060","Text":"Let\u0027s say that 5 over 3 is going to be our common base,"},{"Start":"01:46.060 ","End":"01:47.490","Text":"just make a note to that,"},{"Start":"01:47.490 ","End":"01:49.395","Text":"5 over 3 is what we want."},{"Start":"01:49.395 ","End":"01:51.600","Text":"What do we do about the 3 over 5?"},{"Start":"01:51.600 ","End":"01:59.963","Text":"I believe I mentioned it that 3 over 5 is the same as 1 over 5 over 3."},{"Start":"01:59.963 ","End":"02:01.800","Text":"You must remember this from fraction days,"},{"Start":"02:01.800 ","End":"02:03.150","Text":"if we take 1 over a fraction,"},{"Start":"02:03.150 ","End":"02:04.560","Text":"it\u0027s like inverting it."},{"Start":"02:04.560 ","End":"02:07.850","Text":"According to this rule with n equals 1,"},{"Start":"02:07.850 ","End":"02:11.010","Text":"this is 5 over 3^minus 1."},{"Start":"02:12.020 ","End":"02:16.640","Text":"Now, we\u0027re ready to put everything in terms of 5 over 3."},{"Start":"02:16.640 ","End":"02:19.880","Text":"You see, you have this 5 over 3 again here."},{"Start":"02:19.880 ","End":"02:27.560","Text":"The left-hand side here is 5 over 3^2,"},{"Start":"02:27.560 ","End":"02:34.580","Text":"I\u0027m talking about this line here and all this to the power of 3 minus x."},{"Start":"02:34.580 ","End":"02:36.380","Text":"On the right-hand side,"},{"Start":"02:36.380 ","End":"02:39.785","Text":"I just developed this a little bit more."},{"Start":"02:39.785 ","End":"02:41.240","Text":"This is equal to,"},{"Start":"02:41.240 ","End":"02:43.775","Text":"I need this space so I\u0027ll push this down."},{"Start":"02:43.775 ","End":"02:49.660","Text":"Back here, 3 over 5 we found out was 5 over 3^minus 1."},{"Start":"02:49.660 ","End":"02:57.540","Text":"This is 5 over 3 to the minus 1 but all this to the power of 3."},{"Start":"02:57.540 ","End":"03:01.020","Text":"I just replaced the 3/5 by this."},{"Start":"03:01.020 ","End":"03:07.425","Text":"Now, I can use the rule for power of a power;"},{"Start":"03:07.425 ","End":"03:09.450","Text":"we just multiply the powers."},{"Start":"03:09.450 ","End":"03:17.010","Text":"It\u0027s 5 over 3^minus 1 times 3 is minus 3."},{"Start":"03:17.010 ","End":"03:19.413","Text":"Now I\u0027m moving back here."},{"Start":"03:19.413 ","End":"03:22.865","Text":"What we have is a set of 27 over 125,"},{"Start":"03:22.865 ","End":"03:24.665","Text":"I\u0027ll put what it\u0027s equal to,"},{"Start":"03:24.665 ","End":"03:29.480","Text":"which is 5 over 3^minus 3,"},{"Start":"03:29.480 ","End":"03:35.000","Text":"and all this to the power of x minus 2."},{"Start":"03:35.000 ","End":"03:38.945","Text":"Now, I\u0027m going to use this one twice,"},{"Start":"03:38.945 ","End":"03:41.465","Text":"on the left-hand side and on the right-hand side,"},{"Start":"03:41.465 ","End":"03:49.785","Text":"we get 5 over 3^2 times 3 minus"},{"Start":"03:49.785 ","End":"03:58.127","Text":"x=5 over 3^minus 3"},{"Start":"03:58.127 ","End":"04:00.295","Text":"times x minus 2."},{"Start":"04:00.295 ","End":"04:03.545","Text":"This time we have an exponent with the same base."},{"Start":"04:03.545 ","End":"04:09.305","Text":"We can compare the exponents and say twice"},{"Start":"04:09.305 ","End":"04:17.340","Text":"3 minus x is equal to minus 3x minus 2."},{"Start":"04:17.340 ","End":"04:19.560","Text":"All this is getting in the way."},{"Start":"04:19.560 ","End":"04:22.474","Text":"Let\u0027s continue over here."},{"Start":"04:22.474 ","End":"04:26.625","Text":"We expand brackets and get 2 times 3 is 6,"},{"Start":"04:26.625 ","End":"04:32.290","Text":"minus 2x equals minus 3x,"},{"Start":"04:32.290 ","End":"04:35.773","Text":"minus 3 times minus 2 is plus 6."},{"Start":"04:35.773 ","End":"04:40.920","Text":"Bring the x\u0027s to the left and numbers to the right,"},{"Start":"04:40.920 ","End":"04:48.815","Text":"I have minus 2x plus 3x equals 6 minus 6."},{"Start":"04:48.815 ","End":"04:51.110","Text":"This gives me x,"},{"Start":"04:51.110 ","End":"04:53.225","Text":"this gives me 0,"},{"Start":"04:53.225 ","End":"04:55.775","Text":"looks like this is the answer."},{"Start":"04:55.775 ","End":"04:58.025","Text":"Tell you what? Let\u0027s check it."},{"Start":"04:58.025 ","End":"05:03.770","Text":"Let\u0027s substitute x=0 in the original question and see what we get."},{"Start":"05:03.770 ","End":"05:07.095","Text":"Let\u0027s see what the left-hand side is."},{"Start":"05:07.095 ","End":"05:08.970","Text":"That\u0027s one way of checking equality,"},{"Start":"05:08.970 ","End":"05:10.100","Text":"check the left-hand side,"},{"Start":"05:10.100 ","End":"05:13.895","Text":"check the right-hand side and see that it comes to the same thing."},{"Start":"05:13.895 ","End":"05:21.285","Text":"On the left, we have 25 over 9^3 minus x is 3 minus 0,"},{"Start":"05:21.285 ","End":"05:28.450","Text":"this is equal to 25 over 9^3."},{"Start":"05:28.450 ","End":"05:35.284","Text":"I\u0027ll save the step by putting a 3 at the top and on the bottom."},{"Start":"05:35.284 ","End":"05:41.689","Text":"There I\u0027m talking about this formula and actually in the other direction."},{"Start":"05:41.689 ","End":"05:44.480","Text":"I need a calculator here,"},{"Start":"05:44.480 ","End":"05:53.935","Text":"and I get 15,625 over 729,"},{"Start":"05:53.935 ","End":"05:56.135","Text":"that\u0027s the left-hand side."},{"Start":"05:56.135 ","End":"05:58.310","Text":"Now let\u0027s try the right-hand side."},{"Start":"05:58.310 ","End":"06:02.420","Text":"Here we have 27 over"},{"Start":"06:02.420 ","End":"06:09.615","Text":"125^x minus 2 is 0 minus 2."},{"Start":"06:09.615 ","End":"06:15.290","Text":"You can basically invert the fraction and reverse the sign,"},{"Start":"06:15.290 ","End":"06:21.700","Text":"125 over 27^plus 2."},{"Start":"06:21.700 ","End":"06:25.505","Text":"Again on the calculator, I check this."},{"Start":"06:25.505 ","End":"06:32.495","Text":"This comes out to be 15,625 or 15,625,"},{"Start":"06:32.495 ","End":"06:38.700","Text":"and 27 times 27 does indeed come out to be 729."},{"Start":"06:38.700 ","End":"06:42.020","Text":"The left-hand side equals the right-hand side."},{"Start":"06:42.020 ","End":"06:46.635","Text":"Which means that x equals 0 is a verified solution."},{"Start":"06:46.635 ","End":"06:48.870","Text":"That\u0027s enough with the first one,"},{"Start":"06:48.870 ","End":"06:51.115","Text":"let\u0027s get onto the second one."},{"Start":"06:51.115 ","End":"06:53.420","Text":"Here we have 81 over 16,"},{"Start":"06:53.420 ","End":"06:55.820","Text":"here we have 2 over 3."},{"Start":"06:55.820 ","End":"07:00.709","Text":"Just excuse me a second while I erase this."},{"Start":"07:00.709 ","End":"07:02.180","Text":"As I was saying,"},{"Start":"07:02.180 ","End":"07:06.860","Text":"if you remember your exponents for some of the smaller numbers,"},{"Start":"07:06.860 ","End":"07:12.815","Text":"you know that this is 9 times 9 but it\u0027s also 3 times 3 times 3 times 3."},{"Start":"07:12.815 ","End":"07:20.405","Text":"What I\u0027m getting at is this is 3^4 and 16 is 2 times 2 times 2 times 2 is 2^4."},{"Start":"07:20.405 ","End":"07:23.280","Text":"It\u0027s also 4^2 but it\u0027s 2^4."},{"Start":"07:23.280 ","End":"07:27.050","Text":"I\u0027m already seeing that I have a 3"},{"Start":"07:27.050 ","End":"07:31.055","Text":"over 2 here and I\u0027m looking for a 2 over 3, let\u0027s keep going."},{"Start":"07:31.055 ","End":"07:38.620","Text":"Once again to use this formula here to say that this is 3 over 2^4,"},{"Start":"07:38.620 ","End":"07:43.580","Text":"and then I\u0027m going to use the shortcut with the reciprocal that"},{"Start":"07:43.580 ","End":"07:47.675","Text":"basically you can change the order of a fraction,"},{"Start":"07:47.675 ","End":"07:55.350","Text":"make it upside down and like 2/3 as long as you make the exponent reverse sign also."},{"Start":"07:55.350 ","End":"07:56.930","Text":"That\u0027s a rule of thumb."},{"Start":"07:56.930 ","End":"07:59.010","Text":"A fraction to the power of something,"},{"Start":"07:59.010 ","End":"08:03.920","Text":"you can invert the fraction and negate the exponent. There we are."},{"Start":"08:03.920 ","End":"08:09.595","Text":"Now, we\u0027re going to take 2 over 3 as the common exponent."},{"Start":"08:09.595 ","End":"08:10.940","Text":"We have it on the right,"},{"Start":"08:10.940 ","End":"08:13.120","Text":"and now we\u0027ve seen that we have it on the left also."},{"Start":"08:13.120 ","End":"08:14.720","Text":"Let\u0027s rewrite this."},{"Start":"08:14.720 ","End":"08:22.410","Text":"We only need to rewrite the left-hand side as 2/3^minus 4,"},{"Start":"08:22.410 ","End":"08:24.770","Text":"that\u0027s the 81 over 16 bit,"},{"Start":"08:24.770 ","End":"08:28.310","Text":"but all this to the power of minus x."},{"Start":"08:28.310 ","End":"08:30.440","Text":"On the right-hand side,"},{"Start":"08:30.440 ","End":"08:32.600","Text":"we already have it in terms of 2/3,"},{"Start":"08:32.600 ","End":"08:37.460","Text":"so I\u0027ll just have to copy it as 2/3^5 minus x^2."},{"Start":"08:37.460 ","End":"08:40.430","Text":"Now, we\u0027re back to our desirable case where"},{"Start":"08:40.430 ","End":"08:43.370","Text":"we have the same base but different exponents."},{"Start":"08:43.370 ","End":"08:46.520","Text":"I should first multiply out."},{"Start":"08:46.520 ","End":"08:54.995","Text":"This is 2/3^minus 4 times minus x is 4x,"},{"Start":"08:54.995 ","End":"08:57.035","Text":"minus minus is plus,"},{"Start":"08:57.035 ","End":"09:04.330","Text":"and this equals 2/3^5 minus x^2."},{"Start":"09:04.330 ","End":"09:07.260","Text":"This is the point I wanted to do the highlighting."},{"Start":"09:07.260 ","End":"09:08.460","Text":"We say, yes,"},{"Start":"09:08.460 ","End":"09:11.850","Text":"same base, we can compare the exponents."},{"Start":"09:11.850 ","End":"09:21.595","Text":"From here, we can get that 4x is equal to 5 minus x^2."},{"Start":"09:21.595 ","End":"09:25.160","Text":"Let\u0027s bring everything over to the left-hand side."},{"Start":"09:25.160 ","End":"09:29.895","Text":"We get plus x^2 and the 4x was already here,"},{"Start":"09:29.895 ","End":"09:34.770","Text":"and the 5 I bring over as minus 5 and that equals 0."},{"Start":"09:34.770 ","End":"09:38.750","Text":"Now this looks familiar, quadratic equation."},{"Start":"09:38.750 ","End":"09:46.635","Text":"Quadratic equation where a=1, b=4,"},{"Start":"09:46.635 ","End":"09:54.190","Text":"and c= minus 5 and I\u0027m referring to the standard quadratic equation formula,"},{"Start":"09:54.190 ","End":"09:55.725","Text":"I\u0027ll just quickly write it;"},{"Start":"09:55.725 ","End":"10:05.480","Text":"x= minus b plus or minus the square root of b^2 minus 4ac over 2a."},{"Start":"10:05.480 ","End":"10:07.800","Text":"You should memorize this,"},{"Start":"10:07.800 ","End":"10:11.870","Text":"it\u0027s very useful for quadratic equations at least."},{"Start":"10:11.870 ","End":"10:14.990","Text":"There\u0027s no relation between the a and b here on"},{"Start":"10:14.990 ","End":"10:17.810","Text":"the a and b in this chart, something totally else."},{"Start":"10:17.810 ","End":"10:19.730","Text":"Substituting this here,"},{"Start":"10:19.730 ","End":"10:24.555","Text":"we get that this is equal in our case too, minus 4,"},{"Start":"10:24.555 ","End":"10:33.770","Text":"plus or minus the square root of b^2 minus 4 times a which is 1,"},{"Start":"10:33.770 ","End":"10:37.300","Text":"times c which is minus 5,"},{"Start":"10:37.300 ","End":"10:42.030","Text":"and this all over 2a which is 2 times 1."},{"Start":"10:42.030 ","End":"10:43.860","Text":"I need a bit more space."},{"Start":"10:43.860 ","End":"10:47.505","Text":"It shouldn\u0027t be too hard to do."},{"Start":"10:47.505 ","End":"10:51.995","Text":"This is minus 4 plus or minus the square root."},{"Start":"10:51.995 ","End":"10:54.290","Text":"Let\u0027s see if we can see what\u0027s under the square root sign."},{"Start":"10:54.290 ","End":"10:56.510","Text":"This is 16 on this,"},{"Start":"10:56.510 ","End":"10:57.680","Text":"there\u0027s 2 minuses here,"},{"Start":"10:57.680 ","End":"10:59.180","Text":"so it\u0027s plus 20,"},{"Start":"10:59.180 ","End":"11:07.925","Text":"16 plus 20 is 36 over 2 and now we split up."},{"Start":"11:07.925 ","End":"11:10.655","Text":"The square root of 36 is 6,"},{"Start":"11:10.655 ","End":"11:19.635","Text":"we need minus 4 plus 6 over 2 and minus 4 minus 6 over 2."},{"Start":"11:19.635 ","End":"11:24.690","Text":"This one comes out to be plus 2 over 2 which is 1,"},{"Start":"11:24.690 ","End":"11:29.700","Text":"and this is minus 10 over 2 which is minus 5."},{"Start":"11:29.700 ","End":"11:33.720","Text":"In this case, we actually have two answers."},{"Start":"11:33.720 ","End":"11:40.425","Text":"We have that x could equal 1 and x could equal minus 5,"},{"Start":"11:40.425 ","End":"11:43.365","Text":"both should work if you substitute them."},{"Start":"11:43.365 ","End":"11:46.960","Text":"We are done."}],"ID":8116},{"Watched":false,"Name":"Exercise 4","Duration":"10m 25s","ChapterTopicVideoID":8024,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8024.jpeg","UploadDate":"2020-09-30T14:38:24.2800000","DurationForVideoObject":"PT10M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.155","Text":"Here we have another couple of exponential equations to solve."},{"Start":"00:04.155 ","End":"00:06.510","Text":"We\u0027ll start with the first."},{"Start":"00:06.510 ","End":"00:09.720","Text":"If I look at it, it looks a bit of a mess,"},{"Start":"00:09.720 ","End":"00:10.950","Text":"at least on the left-hand side,"},{"Start":"00:10.950 ","End":"00:12.660","Text":"but if you look closer,"},{"Start":"00:12.660 ","End":"00:16.800","Text":"you\u0027ll see that everything involves 2."},{"Start":"00:16.800 ","End":"00:20.205","Text":"Here we have a base 2 in this exponent,"},{"Start":"00:20.205 ","End":"00:22.320","Text":"8 is 2^3,"},{"Start":"00:22.320 ","End":"00:26.595","Text":"0.25 is just 1/4."},{"Start":"00:26.595 ","End":"00:28.860","Text":"Very well-known decimal."},{"Start":"00:28.860 ","End":"00:34.380","Text":"That\u0027s a quarter and 4 is power of 2 and even on the right-hand side,"},{"Start":"00:34.380 ","End":"00:36.075","Text":"16 is a power of 2."},{"Start":"00:36.075 ","End":"00:39.915","Text":"It looks like we could put everything in terms of base 2."},{"Start":"00:39.915 ","End":"00:46.530","Text":"Let\u0027s see, if I rewrite this 8, this is 2^3."},{"Start":"00:46.530 ","End":"00:48.690","Text":"Here I have 2^x,"},{"Start":"00:48.690 ","End":"00:50.880","Text":"so I\u0027ll just leave it as 2^x."},{"Start":"00:50.880 ","End":"00:53.385","Text":"Now, 0.25,"},{"Start":"00:53.385 ","End":"00:55.350","Text":"let me do this side,"},{"Start":"00:55.350 ","End":"01:05.370","Text":"0.25 is a quarter and a 1/4 is 1 over 2^2."},{"Start":"01:05.370 ","End":"01:07.430","Text":"This is equal to,"},{"Start":"01:07.430 ","End":"01:11.315","Text":"if you take a 1 over an exponent,"},{"Start":"01:11.315 ","End":"01:14.600","Text":"it\u0027s like taking the negative exponent."},{"Start":"01:14.600 ","End":"01:16.610","Text":"Where is that formula sheet?"},{"Start":"01:16.610 ","End":"01:20.239","Text":"Here it is, copied it from the previous exercise with all the markings."},{"Start":"01:20.239 ","End":"01:21.680","Text":"Here this is the 1 I mean,"},{"Start":"01:21.680 ","End":"01:26.105","Text":"1/ 2^2 would be 2^-2."},{"Start":"01:26.105 ","End":"01:35.540","Text":"Back here, this is times 2^-2, instead of the 0.25,"},{"Start":"01:35.540 ","End":"01:40.445","Text":"we still have to the power of 2x^2 plus 1,"},{"Start":"01:40.445 ","End":"01:42.380","Text":"that\u0027s on the left-hand side,"},{"Start":"01:42.380 ","End":"01:45.067","Text":"and now 1/16,"},{"Start":"01:45.067 ","End":"01:47.842","Text":"I\u0027ll also do that at the side."},{"Start":"01:47.842 ","End":"01:51.935","Text":"1/16 is 1/2^4."},{"Start":"01:51.935 ","End":"01:53.342","Text":"We\u0027ve seen this before,"},{"Start":"01:53.342 ","End":"01:56.360","Text":"the 16 is 2^4, and once again,"},{"Start":"01:56.360 ","End":"02:01.170","Text":"using this rule, we get that this is 2^ minus 4."},{"Start":"02:01.170 ","End":"02:05.375","Text":"All we\u0027ll have to do is copy it over here, 2^ minus 4."},{"Start":"02:05.375 ","End":"02:08.795","Text":"Now I still have a bit of organizing to do here."},{"Start":"02:08.795 ","End":"02:10.880","Text":"Let me do a couple of things."},{"Start":"02:10.880 ","End":"02:13.415","Text":"I see here I have a quotient,"},{"Start":"02:13.415 ","End":"02:16.460","Text":"division 2^3 /2^x,"},{"Start":"02:16.460 ","End":"02:21.230","Text":"and this looks like this formula here,"},{"Start":"02:21.230 ","End":"02:24.369","Text":"and for some reason we\u0027re doing the all from right to left,"},{"Start":"02:24.369 ","End":"02:25.850","Text":"I mean this one,"},{"Start":"02:25.850 ","End":"02:29.825","Text":"and in this direction with m being 3,"},{"Start":"02:29.825 ","End":"02:31.580","Text":"n being x,"},{"Start":"02:31.580 ","End":"02:33.125","Text":"and a being 2."},{"Start":"02:33.125 ","End":"02:38.210","Text":"This bit I can write as 2 to the power of 3 minus x,"},{"Start":"02:38.210 ","End":"02:40.940","Text":"that\u0027s the m minus n. Here,"},{"Start":"02:40.940 ","End":"02:43.235","Text":"I\u0027m going to use this formula,"},{"Start":"02:43.235 ","End":"02:46.940","Text":"a power of a power, multiply the powers."},{"Start":"02:46.940 ","End":"02:53.270","Text":"This one is 2 to the power of minus 2 times,"},{"Start":"02:53.270 ","End":"02:55.805","Text":"times I can use with a bracket,"},{"Start":"02:55.805 ","End":"03:01.450","Text":"2x^2 plus 1 still equal 2^ minus 4."},{"Start":"03:01.450 ","End":"03:07.430","Text":"I\u0027m going to use this formula also in this direction from right to left,"},{"Start":"03:07.430 ","End":"03:09.620","Text":"because here I have a product,"},{"Start":"03:09.620 ","End":"03:13.575","Text":"this dot is this dot here, and a is 2,"},{"Start":"03:13.575 ","End":"03:17.625","Text":"so this one comes out to be 2 to the power of,"},{"Start":"03:17.625 ","End":"03:19.050","Text":"this plus this,"},{"Start":"03:19.050 ","End":"03:28.845","Text":"3 minus x plus a negative just means minus 2 times 2x^2 plus 1."},{"Start":"03:28.845 ","End":"03:32.260","Text":"Right-hand side is still 2^ minus 4."},{"Start":"03:32.260 ","End":"03:37.265","Text":"Let\u0027s just compute this bit separately to save carrying it around everywhere,"},{"Start":"03:37.265 ","End":"03:40.410","Text":"3 minus x,"},{"Start":"03:40.410 ","End":"03:47.120","Text":"and here I expand minus 2 times 2x^2 is minus 4x^2,"},{"Start":"03:47.120 ","End":"03:50.620","Text":"minus 2 times 1 is minus 2,"},{"Start":"03:50.620 ","End":"03:53.630","Text":"and then collect everything."},{"Start":"03:53.630 ","End":"03:55.490","Text":"Let\u0027s put it in the right order."},{"Start":"03:55.490 ","End":"04:04.680","Text":"minus 4x^2 minus x plus 3 minus 2 plus 1."},{"Start":"04:04.680 ","End":"04:08.850","Text":"Back here, I\u0027ve got 2 to the power of"},{"Start":"04:08.850 ","End":"04:18.655","Text":"minus 4x^2 minus x plus 1 is equal to 2^4."},{"Start":"04:18.655 ","End":"04:21.275","Text":"Now we\u0027re at the situation we want."},{"Start":"04:21.275 ","End":"04:25.235","Text":"We have an exponent with the same base 2 here and here,"},{"Start":"04:25.235 ","End":"04:27.905","Text":"so we can compare the exponents."},{"Start":"04:27.905 ","End":"04:34.380","Text":"We have minus 4x^2 minus x plus 1."},{"Start":"04:34.380 ","End":"04:35.775","Text":"I\u0027m reading of here,"},{"Start":"04:35.775 ","End":"04:38.775","Text":"is equal to minus 4."},{"Start":"04:38.775 ","End":"04:41.464","Text":"Bring the minus 4 over to the left."},{"Start":"04:41.464 ","End":"04:48.410","Text":"I\u0027ve got minus 4x^2 minus x plus 5 = 0."},{"Start":"04:48.410 ","End":"04:53.255","Text":"Now it\u0027s a quadratic x equals minus b."},{"Start":"04:53.255 ","End":"04:54.980","Text":"That\u0027s plus 1,"},{"Start":"04:54.980 ","End":"04:58.885","Text":"plus or minus the square root of b^2,"},{"Start":"04:58.885 ","End":"05:00.310","Text":"which is minus 1^2,"},{"Start":"05:00.310 ","End":"05:03.145","Text":"which is 1, it is 1^2,"},{"Start":"05:03.145 ","End":"05:13.035","Text":"minus 4 ac minus 4 times a is minus 4 times c is 5 and all this,"},{"Start":"05:13.035 ","End":"05:16.975","Text":"over 2a, twice minus 4."},{"Start":"05:16.975 ","End":"05:19.175","Text":"Now let\u0027s see what\u0027s under the square root sign."},{"Start":"05:19.175 ","End":"05:25.175","Text":"4 times 4 times 5 is 16 times 5 is 80,"},{"Start":"05:25.175 ","End":"05:29.545","Text":"but it\u0027s plus 80 plus the 1 is 81,"},{"Start":"05:29.545 ","End":"05:31.970","Text":"the square root of 81, as we all know,"},{"Start":"05:31.970 ","End":"05:36.275","Text":"is 9 so x ="},{"Start":"05:36.275 ","End":"05:43.960","Text":"1 plus or minus 9 over minus 8."},{"Start":"05:43.960 ","End":"05:46.695","Text":"The 2 possibilities we get are,"},{"Start":"05:46.695 ","End":"05:51.060","Text":"1 minus 9 is minus 8,"},{"Start":"05:51.060 ","End":"05:53.620","Text":"minus 8 over minus 8 is 1."},{"Start":"05:53.620 ","End":"05:59.040","Text":"The other possibility is 1 plus 9, which is 10."},{"Start":"05:59.040 ","End":"06:04.480","Text":"10 over minus 8 is minus 10 over 8,"},{"Start":"06:04.490 ","End":"06:07.924","Text":"minus 5 over 4,"},{"Start":"06:07.924 ","End":"06:12.070","Text":"this is minus 1 and a quarter."},{"Start":"06:12.070 ","End":"06:14.495","Text":"These are the solutions for x,"},{"Start":"06:14.495 ","End":"06:20.310","Text":"either 1 or minus 1 and a quarter."},{"Start":"06:20.310 ","End":"06:21.998","Text":"Onto part b."},{"Start":"06:21.998 ","End":"06:26.603","Text":"Here\u0027s a formula sheet reappeared."},{"Start":"06:26.603 ","End":"06:30.620","Text":"If we look at this, we see we have a 5/3 and we have a 3/5."},{"Start":"06:30.620 ","End":"06:34.535","Text":"Obviously, 1 of these 2 is going to be the base I\u0027m going to use."},{"Start":"06:34.535 ","End":"06:36.725","Text":"But if I look at the other side,"},{"Start":"06:36.725 ","End":"06:41.005","Text":"I see that 9/25 is 3^2/ 5^2,"},{"Start":"06:41.005 ","End":"06:45.105","Text":"so I would rather use the 3/5 as the base,"},{"Start":"06:45.105 ","End":"06:47.684","Text":"and then what I would have,"},{"Start":"06:47.684 ","End":"06:51.750","Text":"the 5/3 can be written in terms of 3/5,"},{"Start":"06:51.750 ","End":"06:55.275","Text":"as 3/5 to the minus 1."},{"Start":"06:55.275 ","End":"06:56.670","Text":"We talked about this,"},{"Start":"06:56.670 ","End":"07:03.095","Text":"if you invert a fraction and make the exponent negative or change its sign,"},{"Start":"07:03.095 ","End":"07:04.450","Text":"then we\u0027re okay,"},{"Start":"07:04.450 ","End":"07:07.600","Text":"and as for the 9/25,"},{"Start":"07:07.600 ","End":"07:09.560","Text":"we\u0027ve already seen this thing."},{"Start":"07:09.560 ","End":"07:12.200","Text":"This is 3^2 /5^2,"},{"Start":"07:12.200 ","End":"07:17.480","Text":"which I write immediately as (3/ 5)^2."},{"Start":"07:17.480 ","End":"07:20.000","Text":"That was the use of this formula."},{"Start":"07:20.000 ","End":"07:23.115","Text":"Plugging it in, both of them,"},{"Start":"07:23.115 ","End":"07:27.690","Text":"we get that this is 3/5 to the minus 1,"},{"Start":"07:27.690 ","End":"07:32.035","Text":"to the power of 1 minus x,"},{"Start":"07:32.035 ","End":"07:34.260","Text":"times 3/5 is good,"},{"Start":"07:34.260 ","End":"07:35.400","Text":"this is our base,"},{"Start":"07:35.400 ","End":"07:38.805","Text":"it\u0027s what want to the minus 1/2,"},{"Start":"07:38.805 ","End":"07:48.370","Text":"and here we have 3/5^2 and then to the power of 2x."},{"Start":"07:48.370 ","End":"07:51.755","Text":"Now you should be getting used to some of these formulas."},{"Start":"07:51.755 ","End":"07:54.680","Text":"There\u0027s this formula which says that the power of"},{"Start":"07:54.680 ","End":"07:59.540","Text":"a power is just the product of the powers roughly."},{"Start":"07:59.540 ","End":"08:02.480","Text":"What we do here is 3/5,"},{"Start":"08:02.480 ","End":"08:08.210","Text":"and then we multiply minus 1 times 1 minus x minus 1."},{"Start":"08:08.210 ","End":"08:11.450","Text":"I\u0027ll need the brackets for 1 minus x,"},{"Start":"08:11.450 ","End":"08:17.340","Text":"3/5 to the power of minus 1/2, and then here,"},{"Start":"08:17.340 ","End":"08:19.635","Text":"3/5 to the power of,"},{"Start":"08:19.635 ","End":"08:25.050","Text":"multiply these, 2 times 2x,"},{"Start":"08:25.050 ","End":"08:29.595","Text":"so what we have is now 3/5,"},{"Start":"08:29.595 ","End":"08:31.265","Text":"if I multiply this out,"},{"Start":"08:31.265 ","End":"08:34.775","Text":"it\u0027s minus 1 plus x."},{"Start":"08:34.775 ","End":"08:39.795","Text":"Here I have minus a 1/2 in the exponent,"},{"Start":"08:39.795 ","End":"08:43.255","Text":"and here I have 4x."},{"Start":"08:43.255 ","End":"08:47.255","Text":"Once again, we\u0027re going to be using this formula that"},{"Start":"08:47.255 ","End":"08:50.930","Text":"if I have a product of exponents with the same base,"},{"Start":"08:50.930 ","End":"08:52.820","Text":"I just add the exponents."},{"Start":"08:52.820 ","End":"08:59.240","Text":"What we get is minus 1 plus x minus 1/2,"},{"Start":"08:59.240 ","End":"09:01.175","Text":"I\u0027m adding but it\u0027s a negative,"},{"Start":"09:01.175 ","End":"09:05.615","Text":"is equal to 3/5 to the power 4x,"},{"Start":"09:05.615 ","End":"09:08.270","Text":"and that will finally at the situation of"},{"Start":"09:08.270 ","End":"09:12.425","Text":"standard situation where we have an exponent with the same base."},{"Start":"09:12.425 ","End":"09:15.620","Text":"That same base is 3/5."},{"Start":"09:15.620 ","End":"09:17.750","Text":"Then we compare exponents."},{"Start":"09:17.750 ","End":"09:26.295","Text":"We get that minus 1 plus x minus 1/2 is equal to 4x."},{"Start":"09:26.295 ","End":"09:29.225","Text":"That bring the xs to the left,"},{"Start":"09:29.225 ","End":"09:31.383","Text":"numbers to the right."},{"Start":"09:31.383 ","End":"09:37.775","Text":"X minus 4x is equal to plus 1 plus a 1/2,"},{"Start":"09:37.775 ","End":"09:40.430","Text":"x minus 4x is minus 3x,"},{"Start":"09:40.430 ","End":"09:42.210","Text":"it\u0027s 1x minus 4x,"},{"Start":"09:42.210 ","End":"09:45.390","Text":"and 1 plus a 1/2 is 1 and 1/2."},{"Start":"09:45.390 ","End":"09:49.005","Text":"But 1 and 1/2 I can write as 3/2,"},{"Start":"09:49.005 ","End":"09:53.405","Text":"and then if I divide both sides by 3,"},{"Start":"09:53.405 ","End":"10:01.135","Text":"I get that x =3 over 2 divided by minus 3."},{"Start":"10:01.135 ","End":"10:05.075","Text":"A division is like multiplying by the reciprocal."},{"Start":"10:05.075 ","End":"10:12.740","Text":"This is actually 3/2 times minus 1 over 3,"},{"Start":"10:12.740 ","End":"10:17.095","Text":"the 3 is cancel and we\u0027re left with minus a 1/2,"},{"Start":"10:17.095 ","End":"10:23.723","Text":"and that is the answer to part b. X is minus 1/2."},{"Start":"10:23.723 ","End":"10:26.050","Text":"We\u0027re done."}],"ID":8117},{"Watched":false,"Name":"Exercise 5","Duration":"8m 25s","ChapterTopicVideoID":8027,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8027.jpeg","UploadDate":"2020-09-30T14:41:07.9700000","DurationForVideoObject":"PT8M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this exercise, we\u0027re given a couple of exponential equations to"},{"Start":"00:04.110 ","End":"00:09.180","Text":"solve and we\u0027ll start with the first one here."},{"Start":"00:09.180 ","End":"00:12.660","Text":"The thing to do is to find what base we want to"},{"Start":"00:12.660 ","End":"00:16.500","Text":"work in and we want to get both sides in terms of the same base."},{"Start":"00:16.500 ","End":"00:21.660","Text":"Well, I think it\u0027s clear that because 16 is 2^4th that we\u0027re going to try for base 2."},{"Start":"00:21.660 ","End":"00:30.615","Text":"What we need to do is to write each of the bases that are currently there in terms of 2."},{"Start":"00:30.615 ","End":"00:33.285","Text":"On the left side, we have 1 over the square root of 2."},{"Start":"00:33.285 ","End":"00:35.415","Text":"On the right-hand side, we have 16."},{"Start":"00:35.415 ","End":"00:38.055","Text":"Like I said, 16 we know is 2^4th,"},{"Start":"00:38.055 ","End":"00:40.255","Text":"1 over square root of 2."},{"Start":"00:40.255 ","End":"00:42.740","Text":"Perhaps we could use our formula sheet."},{"Start":"00:42.740 ","End":"00:45.110","Text":"I think this one will be very useful,"},{"Start":"00:45.110 ","End":"00:47.230","Text":"so will this one."},{"Start":"00:47.230 ","End":"00:49.665","Text":"What we get, first of all,"},{"Start":"00:49.665 ","End":"00:54.600","Text":"1 over 2 to the power of 1/2 using this formula for the square root."},{"Start":"00:54.600 ","End":"00:57.155","Text":"Then the reciprocal, if I read it this way,"},{"Start":"00:57.155 ","End":"01:02.300","Text":"becomes a negative so it\u0027s 2 to the power of minus 1/2."},{"Start":"01:02.300 ","End":"01:05.900","Text":"Now we have both of these things we just substitute in here."},{"Start":"01:05.900 ","End":"01:11.370","Text":"We have 2 to the power of minus 1/2 to the power of"},{"Start":"01:11.370 ","End":"01:19.965","Text":"3 minus x is equal 2 to the power of 4 to the power of minus x."},{"Start":"01:19.965 ","End":"01:24.080","Text":"Now we\u0027re going to use this formula but we\u0027re going to use it from right to left,"},{"Start":"01:24.080 ","End":"01:28.940","Text":"which says that an exponent of an exponent is just like the product of the exponents."},{"Start":"01:28.940 ","End":"01:30.530","Text":"On the left-hand side,"},{"Start":"01:30.530 ","End":"01:33.380","Text":"this is m and this is n. On the right-hand side,"},{"Start":"01:33.380 ","End":"01:38.375","Text":"this is m and this is n. Here we get 2 to the power of the product minus 1/2"},{"Start":"01:38.375 ","End":"01:45.335","Text":"times 3 minus x and here 2 to the power also the product 4 times minus x."},{"Start":"01:45.335 ","End":"01:52.500","Text":"Just multiplying out, we get 2 to the power of minus 1/2 times 3 is minus"},{"Start":"01:52.500 ","End":"02:01.125","Text":"3 over 2 minus 1/2 times minus x is plus 1/2x or x over 2."},{"Start":"02:01.125 ","End":"02:03.460","Text":"This is equal 2."},{"Start":"02:03.460 ","End":"02:06.840","Text":"2 to the power of minus 4x."},{"Start":"02:06.840 ","End":"02:12.005","Text":"Now we have a good situation where we have the same base,"},{"Start":"02:12.005 ","End":"02:15.155","Text":"base 2 here and base 2 here."},{"Start":"02:15.155 ","End":"02:23.285","Text":"That means that we can compare the exponents and now we get minus 3 over 2 plus,"},{"Start":"02:23.285 ","End":"02:27.710","Text":"I\u0027ll write it as 1/2x after all is equal to minus 4x."},{"Start":"02:27.710 ","End":"02:30.350","Text":"Now, as usual with linear equations,"},{"Start":"02:30.350 ","End":"02:32.510","Text":"x is on the left, numbers on the right."},{"Start":"02:32.510 ","End":"02:36.910","Text":"On the left, we have 1/2x plus 4x."},{"Start":"02:36.910 ","End":"02:43.919","Text":"On the right, we have plus 3 over 2 so we get 4 1/2 x."},{"Start":"02:43.919 ","End":"02:51.795","Text":"Well, 4 1/2 you can do in your heads is 9 over 2 times x is 3 over 2."},{"Start":"02:51.795 ","End":"02:56.235","Text":"Now we divide both sides by 9 over 2."},{"Start":"02:56.235 ","End":"02:58.970","Text":"Remember that division is like multiplying"},{"Start":"02:58.970 ","End":"03:01.790","Text":"by the reciprocal so instead of dividing by 9 over 2,"},{"Start":"03:01.790 ","End":"03:04.325","Text":"I multiply by 2 over 9."},{"Start":"03:04.325 ","End":"03:07.040","Text":"Now look, stuff cancels,"},{"Start":"03:07.040 ","End":"03:10.095","Text":"2 with 2 cancels."},{"Start":"03:10.095 ","End":"03:14.325","Text":"I should really write a 1 here and 3 into 9."},{"Start":"03:14.325 ","End":"03:15.540","Text":"They both divide by 3,"},{"Start":"03:15.540 ","End":"03:17.715","Text":"here\u0027s 1, here\u0027s 3,"},{"Start":"03:17.715 ","End":"03:26.625","Text":"and that leaves us with 1 over 3 and that\u0027s our answer for x for this part."},{"Start":"03:26.625 ","End":"03:28.530","Text":"On to part b."},{"Start":"03:28.530 ","End":"03:30.950","Text":"In part b, we also have to choose a base,"},{"Start":"03:30.950 ","End":"03:35.180","Text":"and I think it\u0027s fairly clear that we want to go with base 3 because we"},{"Start":"03:35.180 ","End":"03:39.960","Text":"know that the square root of 3 is 3 to the power of 1/2,"},{"Start":"03:39.960 ","End":"03:42.105","Text":"referring to this formula."},{"Start":"03:42.105 ","End":"03:46.170","Text":"We know that 81 is 3 to the power of 4."},{"Start":"03:46.170 ","End":"03:48.075","Text":"I\u0027ve seen that a lot of times."},{"Start":"03:48.075 ","End":"03:50.975","Text":"We can put everything in terms of 3."},{"Start":"03:50.975 ","End":"03:58.215","Text":"On the left, 3 to the power of 1/2 to the power of 3 minus x and on the right,"},{"Start":"03:58.215 ","End":"04:04.240","Text":"3 to the power of 4 to the power of 1/2x plus 1."},{"Start":"04:06.020 ","End":"04:09.605","Text":"Now we\u0027re going to use this formula."},{"Start":"04:09.605 ","End":"04:11.020","Text":"We should be used to it by now,"},{"Start":"04:11.020 ","End":"04:12.574","Text":"a power of a power,"},{"Start":"04:12.574 ","End":"04:14.600","Text":"then you multiply the powers."},{"Start":"04:14.600 ","End":"04:22.335","Text":"We have 3 to the power of 1/2 times 3 minus x and here,"},{"Start":"04:22.335 ","End":"04:27.745","Text":"3 to the power of 4 times 1/2x plus 1."},{"Start":"04:27.745 ","End":"04:30.140","Text":"This is the good point which we have"},{"Start":"04:30.140 ","End":"04:35.145","Text":"the same base and then we just compare the exponents."},{"Start":"04:35.145 ","End":"04:37.940","Text":"Over here, I\u0027ll do that. Let\u0027s see."},{"Start":"04:37.940 ","End":"04:47.855","Text":"We have 1/2 3 minus x equals 4 times 1/2x plus 1."},{"Start":"04:47.855 ","End":"04:52.555","Text":"Expand brackets, 3 over 2"},{"Start":"04:52.555 ","End":"04:59.410","Text":"minus 1/2x equals 4 times 1/2x,"},{"Start":"04:59.410 ","End":"05:00.960","Text":"4 times 1/2 is 2,"},{"Start":"05:00.960 ","End":"05:05.280","Text":"so it\u0027s 2x plus 4."},{"Start":"05:05.280 ","End":"05:09.675","Text":"I decided I think I\u0027d rather have 3 over 2 as 1 1/2."},{"Start":"05:09.675 ","End":"05:13.175","Text":"No big difference, I just think it\u0027ll be easier to add or subtract."},{"Start":"05:13.175 ","End":"05:15.800","Text":"So x is on the left,"},{"Start":"05:15.800 ","End":"05:18.065","Text":"and I give myself a bit more room here."},{"Start":"05:18.065 ","End":"05:22.135","Text":"We have the minus 1/2x was here,"},{"Start":"05:22.135 ","End":"05:26.220","Text":"2x comes over as minus 2x and on the right,"},{"Start":"05:26.220 ","End":"05:31.550","Text":"4 was there and the minus 1 1/2 comes over that way."},{"Start":"05:31.550 ","End":"05:33.140","Text":"When we have 2 minuses,"},{"Start":"05:33.140 ","End":"05:34.400","Text":"they strengthen each other."},{"Start":"05:34.400 ","End":"05:35.660","Text":"How many x do I have?"},{"Start":"05:35.660 ","End":"05:39.555","Text":"I have 2 plus 1/2 but minus."},{"Start":"05:39.555 ","End":"05:47.025","Text":"It\u0027s minus 2 1/2x and 4 minus 1/2 is also 2 1/2."},{"Start":"05:47.025 ","End":"05:51.215","Text":"If I divide both sides by minus 2 1/2,"},{"Start":"05:51.215 ","End":"05:53.570","Text":"the 2 1/2 cancel, I\u0027m just left with the minus."},{"Start":"05:53.570 ","End":"05:59.540","Text":"Basically, I get minus 1 and that\u0027s the answer for Part B."},{"Start":"05:59.540 ","End":"06:02.755","Text":"The consistency, I circle that."},{"Start":"06:02.755 ","End":"06:05.840","Text":"Why don\u0027t we take the opportunity to check?"},{"Start":"06:05.840 ","End":"06:09.455","Text":"Once in a while, we\u0027ll check the solution by substitution."},{"Start":"06:09.455 ","End":"06:11.945","Text":"I\u0027m going to put x equals minus 1."},{"Start":"06:11.945 ","End":"06:13.250","Text":"In the original equation,"},{"Start":"06:13.250 ","End":"06:16.240","Text":"I have x over here and I have x over here."},{"Start":"06:16.240 ","End":"06:22.190","Text":"Let\u0027s see, left hand side will be the square root of 3 to the power"},{"Start":"06:22.190 ","End":"06:28.185","Text":"of 3 minus and x is minus 1 and I have to check."},{"Start":"06:28.185 ","End":"06:30.420","Text":"I don\u0027t know if it is, I hope it is,"},{"Start":"06:30.420 ","End":"06:36.990","Text":"equal to 81 to the power of 0.5."},{"Start":"06:36.990 ","End":"06:43.275","Text":"I will make that 1/2 times minus 1 and then plus 1."},{"Start":"06:43.275 ","End":"06:45.030","Text":"I\u0027ll tell you what we\u0027ll do,"},{"Start":"06:45.030 ","End":"06:47.930","Text":"we\u0027ll expand the left-hand side to the end and"},{"Start":"06:47.930 ","End":"06:51.365","Text":"then the right-hand side to the end and see if we get equal numbers."},{"Start":"06:51.365 ","End":"06:56.160","Text":"Left-hand side, this is the square root of 3 to the power"},{"Start":"06:56.160 ","End":"07:00.780","Text":"of 3 minus minus 1 is to the power of 4."},{"Start":"07:00.780 ","End":"07:03.725","Text":"Remember the square root is the power of 1/2."},{"Start":"07:03.725 ","End":"07:05.720","Text":"We already used that,"},{"Start":"07:05.720 ","End":"07:07.085","Text":"even have it written here."},{"Start":"07:07.085 ","End":"07:09.230","Text":"Square root is our over 1/2."},{"Start":"07:09.230 ","End":"07:15.805","Text":"This is equal to 3 to the power of 1/2 to the power of 4."},{"Start":"07:15.805 ","End":"07:18.845","Text":"Now, without pulling out the formula,"},{"Start":"07:18.845 ","End":"07:21.980","Text":"we\u0027ve already known that the power of a power in this situation,"},{"Start":"07:21.980 ","End":"07:23.390","Text":"we multiply the powers,"},{"Start":"07:23.390 ","End":"07:27.970","Text":"so it\u0027s 3 to the power of 1/2 times 4."},{"Start":"07:27.970 ","End":"07:35.915","Text":"This means that S is equal to 3 to the power of 2 and ultimately,"},{"Start":"07:35.915 ","End":"07:39.005","Text":"this is equal to 9."},{"Start":"07:39.005 ","End":"07:44.135","Text":"This equals this equals this equals this equals this equals 9."},{"Start":"07:44.135 ","End":"07:47.365","Text":"What about the right-hand side?"},{"Start":"07:47.365 ","End":"07:50.570","Text":"81 to the power of this?"},{"Start":"07:50.570 ","End":"07:52.040","Text":"Let\u0027s see what this is."},{"Start":"07:52.040 ","End":"08:00.240","Text":"It\u0027s 81 to the power of minus 1/2 plus 1 is plus 1/2 and equals this from here."},{"Start":"08:00.240 ","End":"08:02.525","Text":"This is the question mark equal."},{"Start":"08:02.525 ","End":"08:05.420","Text":"This is equal to 81 to the 1/2 and this is equal to,"},{"Start":"08:05.420 ","End":"08:08.345","Text":"we already said the power of 1/2 is the square root,"},{"Start":"08:08.345 ","End":"08:10.295","Text":"the square root of 81,"},{"Start":"08:10.295 ","End":"08:15.115","Text":"and the square root of 81 is 9 because 9 times 9 is 81."},{"Start":"08:15.115 ","End":"08:17.240","Text":"This is equal to this."},{"Start":"08:17.240 ","End":"08:25.660","Text":"So yes, we have confirmed that minus 1 is indeed a solution by substitution. We\u0027re done."}],"ID":8120},{"Watched":false,"Name":"Exercise 6","Duration":"6m 33s","ChapterTopicVideoID":8028,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8028.jpeg","UploadDate":"2020-09-30T14:43:00.7500000","DurationForVideoObject":"PT6M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"Here, we have another couple of exponential equations to solve."},{"Start":"00:03.750 ","End":"00:05.040","Text":"Let\u0027s start with the first,"},{"Start":"00:05.040 ","End":"00:06.570","Text":"then move it up a bit."},{"Start":"00:06.570 ","End":"00:08.970","Text":"The main thing is to decide what\u0027s going to be"},{"Start":"00:08.970 ","End":"00:12.390","Text":"our base and then we want it to be the common base on both sides."},{"Start":"00:12.390 ","End":"00:15.090","Text":"Looking at the numbers here and I see 25,"},{"Start":"00:15.090 ","End":"00:16.510","Text":"I see square root of 5,"},{"Start":"00:16.510 ","End":"00:18.725","Text":"I see 0.2, which is a fifth."},{"Start":"00:18.725 ","End":"00:21.750","Text":"I know I\u0027m going to want to do everything in base 5."},{"Start":"00:21.750 ","End":"00:24.885","Text":"Let\u0027s see if we can rewrite things with 5."},{"Start":"00:24.885 ","End":"00:29.100","Text":"I\u0027ll start with the right-hand side for a change 0.2,"},{"Start":"00:29.100 ","End":"00:32.160","Text":"I want to write that as somehow involving 5."},{"Start":"00:32.160 ","End":"00:34.830","Text":"This is equal to 1/5,"},{"Start":"00:34.830 ","End":"00:37.934","Text":"and 1/5, I don\u0027t even have to bring out the formula."},{"Start":"00:37.934 ","End":"00:41.810","Text":"Remember, 1 over something means to the power of minus 1."},{"Start":"00:41.810 ","End":"00:45.405","Text":"That already gives me as an exponent of 5."},{"Start":"00:45.405 ","End":"00:47.590","Text":"Now how about the left?"},{"Start":"00:47.620 ","End":"00:53.362","Text":"1/25 root 5 is, 1 over,"},{"Start":"00:53.362 ","End":"00:55.805","Text":"25 is 5^2,"},{"Start":"00:55.805 ","End":"00:59.585","Text":"and square root of 5 is 5^1/2."},{"Start":"00:59.585 ","End":"01:02.630","Text":"I\u0027ll bring the formulas just in case you need them,"},{"Start":"01:02.630 ","End":"01:10.970","Text":"I use this formula for getting the 5^-1 and I used the formula here for getting"},{"Start":"01:10.970 ","End":"01:15.260","Text":"a square root of 5 and now I\u0027m going to use this 1"},{"Start":"01:15.260 ","End":"01:20.945","Text":"here in direction from right to left for doing the denominator here."},{"Start":"01:20.945 ","End":"01:25.745","Text":"We see that it\u0027s 1 over to the 5^2 plus 1/2,"},{"Start":"01:25.745 ","End":"01:27.815","Text":"it\u0027s the m plus n,"},{"Start":"01:27.815 ","End":"01:36.350","Text":"so that\u0027s 5^2.5 and the 1 over I use this formula again, it\u0027s 5^-2.5."},{"Start":"01:36.350 ","End":"01:40.475","Text":"We\u0027ll see if I\u0027ll leave it like this or write it as 5/2,"},{"Start":"01:40.475 ","End":"01:42.185","Text":"we\u0027ll decide as needed."},{"Start":"01:42.185 ","End":"01:46.470","Text":"Now I\u0027ve got both of these in terms of exponents of 5,"},{"Start":"01:46.470 ","End":"01:48.645","Text":"put them in to,"},{"Start":"01:48.645 ","End":"01:50.070","Text":"substitute them I mean,"},{"Start":"01:50.070 ","End":"01:59.910","Text":"and so we get (5^-2.5)^-x equals,"},{"Start":"01:59.910 ","End":"02:08.200","Text":"0.2 is (5^-1)^1.5 x-2."},{"Start":"02:08.200 ","End":"02:13.580","Text":"Now as usual, we use this formula for an exponent of an exponent."},{"Start":"02:13.580 ","End":"02:15.530","Text":"We have to multiply the exponents,"},{"Start":"02:15.530 ","End":"02:21.155","Text":"so it\u0027s (5^-2.5)-x. I\u0027ll write it straightaway is"},{"Start":"02:21.155 ","End":"02:28.220","Text":"2.5 x and here I have 5^-1 times all this."},{"Start":"02:28.220 ","End":"02:35.750","Text":"I\u0027ll do it already as minus a 1/2x and -1(-2) is plus 2."},{"Start":"02:35.750 ","End":"02:40.750","Text":"This is a good situation where we have exponents with the same base."},{"Start":"02:40.750 ","End":"02:42.725","Text":"That base is 5 of course."},{"Start":"02:42.725 ","End":"02:44.900","Text":"At this point we compare the exponents,"},{"Start":"02:44.900 ","End":"02:51.160","Text":"so I get 2.5x=-1/2x plus 2."},{"Start":"02:51.160 ","End":"02:55.125","Text":"Continue over here, bring the x\u0027s to the left, numbers to the right."},{"Start":"02:55.125 ","End":"03:00.330","Text":"2.5x plus 1.5x = 2,"},{"Start":"03:00.330 ","End":"03:02.640","Text":"and 2.5 and 0.5 is 3,"},{"Start":"03:02.640 ","End":"03:08.580","Text":"so I\u0027ve got 3x= 2, and x= 2/3."},{"Start":"03:08.580 ","End":"03:11.900","Text":"That\u0027s our answer for x in part A."},{"Start":"03:11.900 ","End":"03:14.560","Text":"Let\u0027s move on to part B."},{"Start":"03:14.560 ","End":"03:17.850","Text":"What\u0027s going to be our base in part B?"},{"Start":"03:17.850 ","End":"03:19.425","Text":"Well, I see a 4,"},{"Start":"03:19.425 ","End":"03:20.600","Text":"and what do I see here?"},{"Start":"03:20.600 ","End":"03:24.650","Text":"0.125. It\u0027s a familiar number,"},{"Start":"03:24.650 ","End":"03:26.220","Text":"I know it\u0027s an 1/8,"},{"Start":"03:26.220 ","End":"03:28.500","Text":"like 12.5% is an 1/8."},{"Start":"03:28.500 ","End":"03:30.450","Text":"Just something that 1 knows."},{"Start":"03:30.450 ","End":"03:33.240","Text":"If we have a 4 and we have an 1/8 in some form,"},{"Start":"03:33.240 ","End":"03:35.235","Text":"it looks like 2 is going to be our,"},{"Start":"03:35.235 ","End":"03:37.100","Text":"I was going to say common denominator."},{"Start":"03:37.100 ","End":"03:38.685","Text":"It\u0027s like a common denominator,"},{"Start":"03:38.685 ","End":"03:41.210","Text":"2 is going to be our base for the exponents."},{"Start":"03:41.210 ","End":"03:43.550","Text":"Let\u0027s try and write everything in terms of 2."},{"Start":"03:43.550 ","End":"03:51.875","Text":"4 is 2^2 and 0.125,"},{"Start":"03:51.875 ","End":"03:56.595","Text":"which is an 1/8 is 1 over 2^3."},{"Start":"03:56.595 ","End":"03:58.815","Text":"8 is 2 times 2 times 2."},{"Start":"03:58.815 ","End":"04:01.400","Text":"Remember that in the denominator,"},{"Start":"04:01.400 ","End":"04:05.495","Text":"you can take an exponent and make it minus instead of,"},{"Start":"04:05.495 ","End":"04:09.400","Text":"you know, what I mean to, 2^-3 is what I mean."},{"Start":"04:09.400 ","End":"04:11.660","Text":"Now going back here,"},{"Start":"04:11.660 ","End":"04:16.350","Text":"I have the cube root of 2^2,"},{"Start":"04:16.350 ","End":"04:20.930","Text":"that\u0027s the 4^x is equal to"},{"Start":"04:20.930 ","End":"04:29.065","Text":"the 6th root of this which is 2^-3."},{"Start":"04:29.065 ","End":"04:32.115","Text":"Now, I think I\u0027m going to need that formula sheet."},{"Start":"04:32.115 ","End":"04:35.540","Text":"Here we are, and I think we\u0027ll start clean."},{"Start":"04:35.540 ","End":"04:38.405","Text":"What I need this time is,"},{"Start":"04:38.405 ","End":"04:42.515","Text":"you see I have a cube root and I have a 6th root and I notice that in general,"},{"Start":"04:42.515 ","End":"04:44.120","Text":"I have a formula for the nth root."},{"Start":"04:44.120 ","End":"04:46.685","Text":"Up to now we\u0027ve only been using square roots."},{"Start":"04:46.685 ","End":"04:49.235","Text":"Let\u0027s use this formula here."},{"Start":"04:49.235 ","End":"04:54.065","Text":"Of course, I still always use this formula for a power of a power."},{"Start":"04:54.065 ","End":"04:58.980","Text":"Here I get 2^2(x) is 2^2x."},{"Start":"04:58.980 ","End":"05:01.035","Text":"That\u0027s from that formula here,"},{"Start":"05:01.035 ","End":"05:02.790","Text":"but the cube root,"},{"Start":"05:02.790 ","End":"05:04.320","Text":"where n is 3,"},{"Start":"05:04.320 ","End":"05:06.100","Text":"means to the power of a 1/3,"},{"Start":"05:06.100 ","End":"05:10.100","Text":"so all this is to the power of 1/3 from the cube root."},{"Start":"05:10.100 ","End":"05:16.200","Text":"On the other side, from the 6th root, I have (2^-3)^1/6."},{"Start":"05:17.020 ","End":"05:22.945","Text":"Once again, I use this formula from right to left for an exponent of an exponent and"},{"Start":"05:22.945 ","End":"05:31.130","Text":"we get (2^2x)1/3 so that\u0027s just 2/3 x,"},{"Start":"05:31.130 ","End":"05:40.470","Text":"on this side I get (2^-3)1/6= -3/6."},{"Start":"05:40.470 ","End":"05:41.730","Text":"Yeah, I know we can cancel it."},{"Start":"05:41.730 ","End":"05:43.635","Text":"It\u0027s going to be minus 1/2."},{"Start":"05:43.635 ","End":"05:46.580","Text":"Just wanted to say already we\u0027re at the situation where we"},{"Start":"05:46.580 ","End":"05:49.490","Text":"have the same base for the exponent,"},{"Start":"05:49.490 ","End":"05:51.920","Text":"that same base is 2 of course."},{"Start":"05:51.920 ","End":"05:58.095","Text":"Now we can compare the exponents and I say 2/3 x equals,"},{"Start":"05:58.095 ","End":"06:01.155","Text":"I can say this is minus 1/2."},{"Start":"06:01.155 ","End":"06:08.685","Text":"Cos3/6 is 1/2. Now x= - 1/2 and instead of dividing by 2/3,"},{"Start":"06:08.685 ","End":"06:11.650","Text":"I\u0027m going to multiply by 3/2."},{"Start":"06:11.650 ","End":"06:13.700","Text":"Remember that trick that are dividing,"},{"Start":"06:13.700 ","End":"06:16.340","Text":"not a trick, it\u0027s what you learned in fractions."},{"Start":"06:16.340 ","End":"06:20.639","Text":"Dividing by a fraction is like multiplying by the inverse fraction,."},{"Start":"06:20.639 ","End":"06:23.960","Text":"What we get is that x equals, let\u0027s see,"},{"Start":"06:23.960 ","End":"06:27.845","Text":"multiply the numerators and multiply the denominators,"},{"Start":"06:27.845 ","End":"06:30.560","Text":"so x= - 3/4,"},{"Start":"06:30.560 ","End":"06:34.560","Text":"and that\u0027s our answer. We\u0027re done."}],"ID":8121},{"Watched":false,"Name":"Exercise 7","Duration":"11m 2s","ChapterTopicVideoID":8029,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8029.jpeg","UploadDate":"2020-09-30T14:44:59.1230000","DurationForVideoObject":"PT11M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"There\u0027s another pair of exponential equations to solve."},{"Start":"00:04.440 ","End":"00:07.695","Text":"They get a little bit more difficult each time."},{"Start":"00:07.695 ","End":"00:10.200","Text":"Let\u0027s start with the first one."},{"Start":"00:10.200 ","End":"00:16.335","Text":"The first thing to do is decide on what\u0027s the base, the common base."},{"Start":"00:16.335 ","End":"00:19.800","Text":"Well, everything smells of 2 here as a 2 here,"},{"Start":"00:19.800 ","End":"00:20.910","Text":"there\u0027s a 2 here."},{"Start":"00:20.910 ","End":"00:22.965","Text":"We know that 16 is 2^4."},{"Start":"00:22.965 ","End":"00:24.915","Text":"We added is 2^5."},{"Start":"00:24.915 ","End":"00:28.185","Text":"We\u0027re going to go with base 2."},{"Start":"00:28.185 ","End":"00:32.220","Text":"Let\u0027s see if we can do that."},{"Start":"00:32.220 ","End":"00:41.925","Text":"I\u0027m going to write this as the square root of 1/32 is 1/2^5."},{"Start":"00:41.925 ","End":"00:49.860","Text":"We know that 2 times 2 times 2 times 2 times 2 is 32.Here we have 1/2,"},{"Start":"00:49.860 ","End":"00:56.715","Text":"which is 2 to the minus 1^6x minus x^2."},{"Start":"00:56.715 ","End":"00:57.980","Text":"In case you\u0027ve forgotten,"},{"Start":"00:57.980 ","End":"01:01.370","Text":"I\u0027ll bring the formula sheet. Here\u0027s the table."},{"Start":"01:01.370 ","End":"01:02.510","Text":"We\u0027ll just use this one."},{"Start":"01:02.510 ","End":"01:07.075","Text":"Meanwhile, when I put the 1/2 is 2 to the minus 1."},{"Start":"01:07.075 ","End":"01:11.105","Text":"I already see I have 2 square roots of obviously going to use this one."},{"Start":"01:11.105 ","End":"01:12.950","Text":"Anyway, let\u0027s get back here."},{"Start":"01:12.950 ","End":"01:15.890","Text":"This equals. Now 16 is 2^4."},{"Start":"01:15.890 ","End":"01:21.505","Text":"We just know that the square root of 2 is going to be 2^1/2,"},{"Start":"01:21.505 ","End":"01:24.210","Text":"which has mentioned square roots."},{"Start":"01:24.210 ","End":"01:29.390","Text":"We can also put the square root here as to the power of 1/2."},{"Start":"01:29.390 ","End":"01:32.090","Text":"Let\u0027s write this again as in fact,"},{"Start":"01:32.090 ","End":"01:34.085","Text":"we\u0027ll use 2 formulas together."},{"Start":"01:34.085 ","End":"01:37.805","Text":"The 1/2^5 makes it 2 to the minus 5,"},{"Start":"01:37.805 ","End":"01:43.070","Text":"and the square root makes it to the power of 1/2. That\u0027s for this one."},{"Start":"01:43.070 ","End":"01:46.625","Text":"Now here we can already use this formula."},{"Start":"01:46.625 ","End":"01:51.160","Text":"That seems to be our favorite formula and we always use it from right to left."},{"Start":"01:51.160 ","End":"01:55.475","Text":"We get 2 power of minus 1 times this."},{"Start":"01:55.475 ","End":"01:57.380","Text":"Just write it as it won\u0027t do."},{"Start":"01:57.380 ","End":"02:01.595","Text":"The multiplication will just indicate that it\u0027s minus 1 times this."},{"Start":"02:01.595 ","End":"02:06.740","Text":"Here we got a chance to use this formula because,"},{"Start":"02:06.740 ","End":"02:11.585","Text":"we have something to the power of 4 times something to the power of a 1/2."},{"Start":"02:11.585 ","End":"02:17.330","Text":"According to this, that would make it 2^4 plus 1/2, which is 4 and 1/2."},{"Start":"02:17.330 ","End":"02:18.965","Text":"Let\u0027s see what else we can do."},{"Start":"02:18.965 ","End":"02:20.735","Text":"Again, using this formula,"},{"Start":"02:20.735 ","End":"02:25.495","Text":"power of a power 2^(minus 5/2)."},{"Start":"02:25.495 ","End":"02:27.510","Text":"Here, 2 to the power."},{"Start":"02:27.510 ","End":"02:28.845","Text":"Let\u0027s do the multiplication."},{"Start":"02:28.845 ","End":"02:33.090","Text":"It\u0027s minus 6x plus x^2."},{"Start":"02:33.090 ","End":"02:35.580","Text":"Same here, 4 and 1/2."},{"Start":"02:35.580 ","End":"02:40.295","Text":"Now again, we\u0027re going to use this formula from right to left,"},{"Start":"02:40.295 ","End":"02:43.720","Text":"where is 2 and then we have this plus this."},{"Start":"02:43.720 ","End":"02:46.475","Text":"We get that 2 to the power of."},{"Start":"02:46.475 ","End":"02:49.070","Text":"I\u0027ll just write it in a different order when I do the sum,"},{"Start":"02:49.070 ","End":"02:54.065","Text":"let me put the x^2 then the minus 6x,"},{"Start":"02:54.065 ","End":"03:02.155","Text":"then minus 5/2=2^4 and 1/2."},{"Start":"03:02.155 ","End":"03:05.330","Text":"Now we finally the place where we wanted to,"},{"Start":"03:05.330 ","End":"03:10.885","Text":"the basic situation where we have an exponent with the same base on both sides."},{"Start":"03:10.885 ","End":"03:14.415","Text":"Of course, I mean 2 is that same base."},{"Start":"03:14.415 ","End":"03:24.080","Text":"We compare the exponents and we get that x^2 minus 6x minus 5/2,"},{"Start":"03:24.080 ","End":"03:27.620","Text":"let\u0027s split it as 2 and 1/2 because we\u0027ve got a mixed number here."},{"Start":"03:27.620 ","End":"03:32.135","Text":"Let\u0027s be consistent is equal to 4 and 1/2."},{"Start":"03:32.135 ","End":"03:35.420","Text":"Now I\u0027ve got a quadratic equation to solve."},{"Start":"03:35.420 ","End":"03:38.330","Text":"I just have to bring this to the other side."},{"Start":"03:38.330 ","End":"03:40.370","Text":"You know what? I\u0027ll continue up here."},{"Start":"03:40.370 ","End":"03:46.015","Text":"We\u0027ll say that x squared minus 6x,"},{"Start":"03:46.015 ","End":"03:54.205","Text":"minus 2 and 1/2 minus 4 and 1/2 is altogether minus 7 is equal to 0."},{"Start":"03:54.205 ","End":"03:58.594","Text":"We know our quadratic formulas by heart, I suppose."},{"Start":"03:58.594 ","End":"04:01.910","Text":"We get that x = minus b,"},{"Start":"04:01.910 ","End":"04:10.650","Text":"which is 6 plus 6 plus or minus the square root of b^2 Is 36."},{"Start":"04:10.650 ","End":"04:13.350","Text":"That\u0027s 6 times 6 minus 4,"},{"Start":"04:13.350 ","End":"04:17.760","Text":"ac is plus 4 times"},{"Start":"04:17.760 ","End":"04:22.970","Text":"1 times 7 because I had 2 minuses and I combine them already into a plus."},{"Start":"04:22.970 ","End":"04:26.150","Text":"We have the minus and we add the minus 7/2,"},{"Start":"04:26.150 ","End":"04:29.440","Text":"which is just 2 because a is 1."},{"Start":"04:29.440 ","End":"04:32.760","Text":"Now let\u0027s see, 4 times 7 is 28,"},{"Start":"04:32.760 ","End":"04:38.235","Text":"28, and 36 is 64."},{"Start":"04:38.235 ","End":"04:45.420","Text":"We have 6 plus or minus the square root of 64/2,"},{"Start":"04:45.430 ","End":"04:53.300","Text":"which is equal to 6 plus or minus 8/2."},{"Start":"04:53.300 ","End":"05:01.295","Text":"Now we split into plus and minus 6 plus 8/2 is 14/2 is 7,"},{"Start":"05:01.295 ","End":"05:06.285","Text":"and 6 minus 8 is minus 2/2 is minus 1."},{"Start":"05:06.285 ","End":"05:08.945","Text":"We have 2 solutions."},{"Start":"05:08.945 ","End":"05:12.680","Text":"Choose a different color than yellow, x=7,"},{"Start":"05:12.680 ","End":"05:18.090","Text":"or x= minus 1 onto the next one."},{"Start":"05:18.090 ","End":"05:20.310","Text":"What do we have here?"},{"Start":"05:20.310 ","End":"05:25.450","Text":"Well, to me everything shouts out 6 to use as the base."},{"Start":"05:25.450 ","End":"05:27.700","Text":"We have 36, which is 6^2,"},{"Start":"05:27.700 ","End":"05:30.820","Text":"and here we have 6 and we have 36 again."},{"Start":"05:30.820 ","End":"05:35.815","Text":"I propose that we try to put everything in terms of basics here."},{"Start":"05:35.815 ","End":"05:37.765","Text":"Let\u0027s do this carefully."},{"Start":"05:37.765 ","End":"05:42.490","Text":"Here we have the x plus 1th root of now,"},{"Start":"05:42.490 ","End":"05:46.040","Text":"36 I\u0027m going to replaced by 6^2."},{"Start":"05:46.040 ","End":"05:48.360","Text":"6 times 6 is 36."},{"Start":"05:48.360 ","End":"05:49.965","Text":"The same thing here,"},{"Start":"05:49.965 ","End":"05:53.360","Text":"0.5x minus 1,"},{"Start":"05:53.360 ","End":"05:58.030","Text":"the denominator, I\u0027m also going to write the 36 as 6^2."},{"Start":"05:58.030 ","End":"05:59.950","Text":"On the right-hand side,"},{"Start":"05:59.950 ","End":"06:05.730","Text":"I\u0027m going to write the square root of 6 as 6^1.5."},{"Start":"06:05.730 ","End":"06:08.570","Text":"That\u0027s from this formula here we use that a lot"},{"Start":"06:08.570 ","End":"06:12.410","Text":"the square root of something means exponent is a 1/2."},{"Start":"06:12.410 ","End":"06:19.660","Text":"What I suggest is multiplying both sides by 6^2 and getting rid of the fraction."},{"Start":"06:19.660 ","End":"06:22.070","Text":"Let me just do the right-hand side first."},{"Start":"06:22.070 ","End":"06:24.905","Text":"Let\u0027s see easiest here I get 6^2,"},{"Start":"06:24.905 ","End":"06:28.460","Text":"6^1/2, and that\u0027s gotten rid of that."},{"Start":"06:28.460 ","End":"06:30.725","Text":"Now I\u0027ve just got the numerator here."},{"Start":"06:30.725 ","End":"06:36.155","Text":"But I also want to use the formula for the roots of general route."},{"Start":"06:36.155 ","End":"06:39.155","Text":"Here I have, instead of n,"},{"Start":"06:39.155 ","End":"06:41.240","Text":"I have x plus 1."},{"Start":"06:41.240 ","End":"06:45.125","Text":"The x plus 1 root means to the power of 1 over."},{"Start":"06:45.125 ","End":"06:50.660","Text":"What I\u0027m saying in short is that if I take what was under the root sign,"},{"Start":"06:50.660 ","End":"06:52.310","Text":"I\u0027ll copy it in a moment."},{"Start":"06:52.310 ","End":"06:58.385","Text":"This will be 1/x plus 1 of the same thing from this formula here."},{"Start":"06:58.385 ","End":"07:03.845","Text":"What we had here was 6^2 to the power of this."},{"Start":"07:03.845 ","End":"07:07.400","Text":"But I also want to expand this using this formula."},{"Start":"07:07.400 ","End":"07:12.910","Text":"It\u0027s 2 times 0.5x minus 1."},{"Start":"07:12.910 ","End":"07:17.030","Text":"This I expand using this formula and the radical,"},{"Start":"07:17.030 ","End":"07:19.850","Text":"the root is using this formula."},{"Start":"07:19.850 ","End":"07:23.464","Text":"This just disappear to the other side because I multiplied."},{"Start":"07:23.464 ","End":"07:27.274","Text":"Continuing. If I do this product,"},{"Start":"07:27.274 ","End":"07:31.265","Text":"I\u0027ve got 6^2 times 0.5 is 1,"},{"Start":"07:31.265 ","End":"07:34.065","Text":"so it\u0027s x minus 2,"},{"Start":"07:34.065 ","End":"07:38.550","Text":"but it\u0027s still to the power of 1/x plus 1."},{"Start":"07:38.550 ","End":"07:41.515","Text":"Here I have 6^2."},{"Start":"07:41.515 ","End":"07:44.300","Text":"I\u0027m using this formula now from right to left,"},{"Start":"07:44.300 ","End":"07:47.480","Text":"I add the exponents when it\u0027s a multiplication here,"},{"Start":"07:47.480 ","End":"07:49.275","Text":"then I add the exponents,"},{"Start":"07:49.275 ","End":"07:53.095","Text":"so it\u0027s 6^2 and 1/2."},{"Start":"07:53.095 ","End":"07:55.550","Text":"I\u0027ll continue over here."},{"Start":"07:55.550 ","End":"07:59.255","Text":"This again is an exponent times an exponent."},{"Start":"07:59.255 ","End":"08:00.995","Text":"It\u0027s this rule here."},{"Start":"08:00.995 ","End":"08:02.920","Text":"This times this."},{"Start":"08:02.920 ","End":"08:05.900","Text":"What I get is 6 to the power of,"},{"Start":"08:05.900 ","End":"08:08.870","Text":"instead of writing x minus 2 times 1 over this,"},{"Start":"08:08.870 ","End":"08:13.295","Text":"I\u0027ll just write it as x minus 2/(x plus 1)."},{"Start":"08:13.295 ","End":"08:14.930","Text":"I want to make a note at the side."},{"Start":"08:14.930 ","End":"08:21.200","Text":"It\u0027s like saying that a times 1/b is a/b, not the same."},{"Start":"08:21.200 ","End":"08:23.690","Text":"A and b is here just in general."},{"Start":"08:23.690 ","End":"08:26.555","Text":"This times, this just means this over this,"},{"Start":"08:26.555 ","End":"08:30.610","Text":"and it\u0027s still equal to 6^ 2.5."},{"Start":"08:30.610 ","End":"08:34.820","Text":"This is the point at which we have an exponent with the same base,"},{"Start":"08:34.820 ","End":"08:37.070","Text":"that base being 6, of course."},{"Start":"08:37.070 ","End":"08:39.515","Text":"Now I just compare the exponents."},{"Start":"08:39.515 ","End":"08:47.510","Text":"I get x minus 2/x plus 1= 2 and 1/2."},{"Start":"08:47.510 ","End":"08:52.235","Text":"Let\u0027s write 2 and 1/2 as 5/2,"},{"Start":"08:52.235 ","End":"08:55.070","Text":"that will be better and we\u0027ll deal with fractions."},{"Start":"08:55.070 ","End":"08:58.130","Text":"Now, here we have an equation with"},{"Start":"08:58.130 ","End":"09:03.560","Text":"fractions and we can find the common denominator and we can multiply out."},{"Start":"09:03.560 ","End":"09:05.150","Text":"I just wanted to make a note at this point."},{"Start":"09:05.150 ","End":"09:06.800","Text":"When we come to the end,"},{"Start":"09:06.800 ","End":"09:13.685","Text":"we have to make sure that our x that we get doesn\u0027t give us a 0 in the denominator."},{"Start":"09:13.685 ","End":"09:15.920","Text":"For those of you who\u0027ve learned what a domain is,"},{"Start":"09:15.920 ","End":"09:18.665","Text":"it means that x is in the domain of this expression."},{"Start":"09:18.665 ","End":"09:22.730","Text":"We can see now that the only thing that would spoil things is if x was minus 1."},{"Start":"09:22.730 ","End":"09:25.100","Text":"We just make sure at the end that x is not minus"},{"Start":"09:25.100 ","End":"09:28.850","Text":"1 because then that will be bad and we\u0027ll have a denominator 0."},{"Start":"09:28.850 ","End":"09:31.825","Text":"What I suggest is the common denominator."},{"Start":"09:31.825 ","End":"09:37.100","Text":"The obvious thing to try is to multiply both sides by twice x plus 1,"},{"Start":"09:37.100 ","End":"09:39.515","Text":"just the product of these 2 denominators."},{"Start":"09:39.515 ","End":"09:41.240","Text":"Then the usual system,"},{"Start":"09:41.240 ","End":"09:44.690","Text":"we see that what we have leftover from x"},{"Start":"09:44.690 ","End":"09:47.985","Text":"plus 1 as we have to have a factor of 2 leftover."},{"Start":"09:47.985 ","End":"09:51.360","Text":"Here we have a factor of x plus 1,"},{"Start":"09:51.360 ","End":"09:53.990","Text":"so we do the multiplication."},{"Start":"09:53.990 ","End":"10:01.290","Text":"We get twice x minus 2=x plus 1 times 5 segments,"},{"Start":"10:01.290 ","End":"10:03.645","Text":"5 times x plus 1."},{"Start":"10:03.645 ","End":"10:07.755","Text":"Bracket expansion 2x minus 4."},{"Start":"10:07.755 ","End":"10:11.025","Text":"Here, 5x plus 5,"},{"Start":"10:11.025 ","End":"10:14.435","Text":"x is to the left numbers to the right, 2x minus 5x."},{"Start":"10:14.435 ","End":"10:17.180","Text":"I can do that in my head minus a 3x."},{"Start":"10:17.180 ","End":"10:21.125","Text":"Here, 5 plus 4 is 9,"},{"Start":"10:21.125 ","End":"10:24.105","Text":"dividing by minus 3,"},{"Start":"10:24.105 ","End":"10:28.690","Text":"we get that x=minus 3."},{"Start":"10:28.690 ","End":"10:33.170","Text":"Like I said, we mentioned that minus 3 is okay here."},{"Start":"10:33.170 ","End":"10:36.440","Text":"Minus 3 plus 1 is not 0 only minus 1."},{"Start":"10:36.440 ","End":"10:38.135","Text":"We didn\u0027t get minus 1."},{"Start":"10:38.135 ","End":"10:40.190","Text":"This is, our answer,"},{"Start":"10:40.190 ","End":"10:43.550","Text":"it would be a good idea to check by substitution,"},{"Start":"10:43.550 ","End":"10:46.490","Text":"but we\u0027ve gone well over time,"},{"Start":"10:46.490 ","End":"10:48.485","Text":"so we\u0027ll skip that."},{"Start":"10:48.485 ","End":"10:50.150","Text":"But if you have the time,"},{"Start":"10:50.150 ","End":"10:53.330","Text":"I recommend substituting and seeing that in"},{"Start":"10:53.330 ","End":"10:57.124","Text":"every exercise and making sure that you get a true statement,"},{"Start":"10:57.124 ","End":"11:00.065","Text":"that it actually works."},{"Start":"11:00.065 ","End":"11:03.450","Text":"I\u0027m stopping here."}],"ID":8122},{"Watched":false,"Name":"Exercise 8","Duration":"11m 38s","ChapterTopicVideoID":8030,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8030.jpeg","UploadDate":"2020-09-30T14:46:09.1830000","DurationForVideoObject":"PT11M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.475","Text":"Here we have another pair of"},{"Start":"00:02.475 ","End":"00:07.740","Text":"exponential equations to solve and we\u0027ll start with the first one."},{"Start":"00:07.740 ","End":"00:14.040","Text":"The first thing we want to do is to find what is the common base going to be."},{"Start":"00:14.040 ","End":"00:15.540","Text":"I think it\u0027s pretty clear."},{"Start":"00:15.540 ","End":"00:16.950","Text":"It\u0027s going to be 3."},{"Start":"00:16.950 ","End":"00:19.860","Text":"Everything is powers of 3 here."},{"Start":"00:19.860 ","End":"00:24.840","Text":"Let\u0027s write this as 27 is 3^3."},{"Start":"00:24.840 ","End":"00:28.755","Text":"We\u0027ve seen this enough times to the minus 1."},{"Start":"00:28.755 ","End":"00:33.050","Text":"Here, we have 3(x)^x."},{"Start":"00:33.050 ","End":"00:35.015","Text":"We can already use the formula."},{"Start":"00:35.015 ","End":"00:36.530","Text":"Here\u0027s my formula sheet."},{"Start":"00:36.530 ","End":"00:40.070","Text":"Yes, this is the one I\u0027m talking about from right to left,"},{"Start":"00:40.070 ","End":"00:43.325","Text":"m and n are both going to be x now is 3,"},{"Start":"00:43.325 ","End":"00:49.810","Text":"3^x times x is 3^x^2."},{"Start":"00:49.810 ","End":"00:55.076","Text":"Then we have 1/9 is 3^2."},{"Start":"00:55.076 ","End":"00:59.060","Text":"It\u0027s just 3^2 to the minus x."},{"Start":"00:59.060 ","End":"01:00.835","Text":"Let\u0027s keep developing this."},{"Start":"01:00.835 ","End":"01:05.240","Text":"Here, I can multiply the exponents once again with this formula."},{"Start":"01:05.240 ","End":"01:09.620","Text":"I\u0027ve got 3^ minus 3 times,"},{"Start":"01:09.620 ","End":"01:14.110","Text":"I don\u0027t need the brackets here, 3^x^2."},{"Start":"01:14.110 ","End":"01:18.305","Text":"Here, I can use the formula."},{"Start":"01:18.305 ","End":"01:23.135","Text":"I can use minus x is n minus n is plus x."},{"Start":"01:23.135 ","End":"01:29.570","Text":"This is 3^2^ positive x in the numerator plus x."},{"Start":"01:29.570 ","End":"01:32.390","Text":"On the left-hand side,"},{"Start":"01:32.390 ","End":"01:34.895","Text":"I can use this formula."},{"Start":"01:34.895 ","End":"01:37.580","Text":"Again from right to left, I can add these 2,"},{"Start":"01:37.580 ","End":"01:44.555","Text":"I get 3^ this plus this minus 3 plus x^2."},{"Start":"01:44.555 ","End":"01:49.505","Text":"Here once again, this formula, it\u0027s 3^2x."},{"Start":"01:49.505 ","End":"01:52.400","Text":"Now we\u0027ve got the ideal situation."},{"Start":"01:52.400 ","End":"01:57.275","Text":"We have an equation of exponents but with the same base 3."},{"Start":"01:57.275 ","End":"02:01.970","Text":"We just throw out the bases and just compare the exponents and say,"},{"Start":"02:01.970 ","End":"02:06.935","Text":"minus 3 plus x^2=2x."},{"Start":"02:06.935 ","End":"02:08.720","Text":"If I bring everything to the left,"},{"Start":"02:08.720 ","End":"02:12.995","Text":"I get a nice quadratic also slightly re-changing the order x^2,"},{"Start":"02:12.995 ","End":"02:17.890","Text":"then the minus 2x and now the minus 3=0."},{"Start":"02:17.890 ","End":"02:22.280","Text":"Nice quadratic, let\u0027s solve it. I\u0027ll do it up here."},{"Start":"02:22.280 ","End":"02:26.630","Text":"We\u0027ve got x is equal to minus b,"},{"Start":"02:26.630 ","End":"02:32.855","Text":"that\u0027s plus 2 plus or minus the square root of b^2,"},{"Start":"02:32.855 ","End":"02:34.620","Text":"is the same as plus 2^2,"},{"Start":"02:34.620 ","End":"02:36.390","Text":"the minus squared is plus,"},{"Start":"02:36.390 ","End":"02:40.485","Text":"minus 4 times a, which is 1,"},{"Start":"02:40.485 ","End":"02:43.740","Text":"times c, which is minus 3,"},{"Start":"02:43.740 ","End":"02:49.435","Text":"all over 2a which is 2 times 1."},{"Start":"02:49.435 ","End":"02:52.025","Text":"Now let\u0027s see what does that leave us with?"},{"Start":"02:52.025 ","End":"02:53.630","Text":"Under the square root sign,"},{"Start":"02:53.630 ","End":"02:58.140","Text":"I have 4 times 3 is 12 and as a minus,"},{"Start":"02:58.140 ","End":"03:02.707","Text":"minus, it\u0027s plus 12 plus 4, which is 16."},{"Start":"03:02.707 ","End":"03:09.395","Text":"I\u0027ll make a note to myself that what I have is scarce the square root of 16, which is 4."},{"Start":"03:09.395 ","End":"03:19.495","Text":"Now, I see that I get that x equals 2 plus or minus 4/2,"},{"Start":"03:19.495 ","End":"03:22.610","Text":"which gives me, if I take the plus sign,"},{"Start":"03:22.610 ","End":"03:24.200","Text":"2 plus 4 is 6,"},{"Start":"03:24.200 ","End":"03:25.685","Text":"over 2 is 3."},{"Start":"03:25.685 ","End":"03:27.205","Text":"If I take minus,"},{"Start":"03:27.205 ","End":"03:31.860","Text":"2 minus 4 is minus 2/2 is minus 1."},{"Start":"03:31.860 ","End":"03:37.275","Text":"These are my two solutions for x, I\u0027ll highlight them."},{"Start":"03:37.275 ","End":"03:40.745","Text":"I think it\u0027s about time to do another one of those,"},{"Start":"03:40.745 ","End":"03:43.550","Text":"checking and verifying the solution. I\u0027ll take one of them."},{"Start":"03:43.550 ","End":"03:46.475","Text":"I\u0027ll verify that 3 really is a solution"},{"Start":"03:46.475 ","End":"03:50.020","Text":"by substituting in here just to get a bit more space there."},{"Start":"03:50.020 ","End":"03:55.630","Text":"I\u0027m checking this one and putting it in here instead of x. I get"},{"Start":"03:55.630 ","End":"04:05.570","Text":"27^minus 1 times 3^3^3 equals,"},{"Start":"04:05.570 ","End":"04:07.190","Text":"but it\u0027s equals question mark,"},{"Start":"04:07.190 ","End":"04:10.472","Text":"because that\u0027s what I want to check."},{"Start":"04:10.472 ","End":"04:14.280","Text":"1/9^minus 3."},{"Start":"04:14.280 ","End":"04:16.995","Text":"Let\u0027s do the left-hand side first."},{"Start":"04:16.995 ","End":"04:22.430","Text":"I\u0027ll call this one left-hand side and this right-hand side,"},{"Start":"04:22.430 ","End":"04:24.545","Text":"sometimes we use these abbreviations."},{"Start":"04:24.545 ","End":"04:30.420","Text":"The left-hand side gives us 27^minus 1,"},{"Start":"04:30.420 ","End":"04:33.585","Text":"I\u0027ll write that straightaway as 1/27,"},{"Start":"04:33.585 ","End":"04:41.130","Text":"and 3^3 is 27^3,"},{"Start":"04:41.130 ","End":"04:44.495","Text":"and this equals, let\u0027s see,"},{"Start":"04:44.495 ","End":"04:46.820","Text":"it\u0027s 27 times 27."},{"Start":"04:46.820 ","End":"04:52.165","Text":"I\u0027ll do it very naively times 27 and it\u0027s over 27."},{"Start":"04:52.165 ","End":"05:01.955","Text":"One of the 27\u0027s cancel and I\u0027m left with 27 times 27, which is 729."},{"Start":"05:01.955 ","End":"05:06.830","Text":"Now, let\u0027s try the right-hand side and see what we get."},{"Start":"05:06.830 ","End":"05:10.820","Text":"Here we have 1/9^minus 3,"},{"Start":"05:10.820 ","End":"05:13.880","Text":"which like 9^plus 3."},{"Start":"05:13.880 ","End":"05:18.392","Text":"I\u0027m just using this formula with n being minus 3,"},{"Start":"05:18.392 ","End":"05:20.955","Text":"minus n is plus 3."},{"Start":"05:20.955 ","End":"05:25.875","Text":"This is equal to 9 times 9 times 9."},{"Start":"05:25.875 ","End":"05:28.245","Text":"If you multiply it out,"},{"Start":"05:28.245 ","End":"05:30.950","Text":"you will get 729."},{"Start":"05:30.950 ","End":"05:35.135","Text":"Perhaps I should have been writing the equals signs."},{"Start":"05:35.135 ","End":"05:40.595","Text":"Anyway, 729=729 and so,"},{"Start":"05:40.595 ","End":"05:43.865","Text":"this solution is verified."},{"Start":"05:43.865 ","End":"05:46.755","Text":"One of them is right, probably the other one is right also."},{"Start":"05:46.755 ","End":"05:53.730","Text":"Let\u0027s move on to Part B. I think I\u0027ll erase this stuff."},{"Start":"05:53.730 ","End":"05:57.075","Text":"That\u0027s better. In Part B,"},{"Start":"05:57.075 ","End":"06:05.540","Text":"it looks like we\u0027re going to use base 2 because we know that 0.5 is 1/2 and 4 is 2^2."},{"Start":"06:05.540 ","End":"06:10.685","Text":"Let\u0027s start working at getting both sides to be 2^something."},{"Start":"06:10.685 ","End":"06:17.820","Text":"This is 1/2^2x minus 4."},{"Start":"06:17.820 ","End":"06:23.910","Text":"On this side I have 1^4 or"},{"Start":"06:23.910 ","End":"06:29.805","Text":"write it as 2^2^x minus 3,"},{"Start":"06:29.805 ","End":"06:33.275","Text":"but still keep it as 8/x."},{"Start":"06:33.275 ","End":"06:35.060","Text":"Notice I don\u0027t touch this 8,"},{"Start":"06:35.060 ","End":"06:40.153","Text":"they don\u0027t say that this 8 is 2^3 or something because it\u0027s an exponent,"},{"Start":"06:40.153 ","End":"06:42.050","Text":"that doesn\u0027t apply to that."},{"Start":"06:42.050 ","End":"06:46.605","Text":"We can write this as 2^minus 1."},{"Start":"06:46.605 ","End":"06:49.940","Text":"I don\u0027t have to refer you all the time to this formula."},{"Start":"06:49.940 ","End":"06:56.480","Text":"But basically you can flip a fraction as long as you negate the exponent,"},{"Start":"06:56.480 ","End":"07:03.520","Text":"so instead of 2^1 is 2^minus 1^2x minus 4."},{"Start":"07:03.520 ","End":"07:05.990","Text":"Here, we can also do the same thing."},{"Start":"07:05.990 ","End":"07:11.485","Text":"We can 1 over means we can make the exponent negative."},{"Start":"07:11.485 ","End":"07:19.520","Text":"We can say that this is 2^2^minus,"},{"Start":"07:19.520 ","End":"07:21.120","Text":"of x minus 3."},{"Start":"07:21.120 ","End":"07:22.500","Text":"How would I write that?"},{"Start":"07:22.500 ","End":"07:26.630","Text":"Let\u0027s write it as minus x minus 3 in the exponent."},{"Start":"07:26.630 ","End":"07:32.170","Text":"But still, all this is ^8/x."},{"Start":"07:32.170 ","End":"07:37.010","Text":"Now, it\u0027s time to start using this equations."},{"Start":"07:37.010 ","End":"07:39.530","Text":"Heavy duty, you can use it a lot."},{"Start":"07:39.530 ","End":"07:46.005","Text":"Here we\u0027ve got 2^minus 1 times 2x minus 4."},{"Start":"07:46.005 ","End":"07:50.120","Text":"Here, we have 2^2 times this."},{"Start":"07:50.120 ","End":"07:52.535","Text":"I can put the minus first, otherwise it\u0027s a mess."},{"Start":"07:52.535 ","End":"07:59.960","Text":"I\u0027ll put the 2, slip it in between the minus and the x minus 3 and all this to ^8/x."},{"Start":"07:59.960 ","End":"08:02.306","Text":"I\u0027ll tell you what, we\u0027re going to use the formula,"},{"Start":"08:02.306 ","End":"08:04.790","Text":"again, we\u0027re going to use it twice in the same row."},{"Start":"08:04.790 ","End":"08:10.370","Text":"I\u0027m going to multiply this also by 8/x and save ourselves the line."},{"Start":"08:10.370 ","End":"08:13.940","Text":"Now [inaudible] the situation that we like,"},{"Start":"08:13.940 ","End":"08:16.220","Text":"we have an exponent with the same base,"},{"Start":"08:16.220 ","End":"08:18.410","Text":"which I mean this 2, of course."},{"Start":"08:18.410 ","End":"08:20.900","Text":"That means I can compare exponents,"},{"Start":"08:20.900 ","End":"08:27.285","Text":"I can say minus 2x minus 4 equals,"},{"Start":"08:27.285 ","End":"08:29.360","Text":"and I can slightly rewrite this,"},{"Start":"08:29.360 ","End":"08:37.320","Text":"I can write minus 2 times 8 is minus 16 times x minus 3."},{"Start":"08:37.320 ","End":"08:40.670","Text":"Let\u0027s take the x and the denominator here."},{"Start":"08:40.670 ","End":"08:44.300","Text":"But really I just compared this and this slightly writing."},{"Start":"08:44.300 ","End":"08:46.235","Text":"Let\u0027s continue over here."},{"Start":"08:46.235 ","End":"08:49.280","Text":"What I want to do is get rid of the fraction by"},{"Start":"08:49.280 ","End":"08:54.010","Text":"multiplying both sides of the equation by x."},{"Start":"08:54.010 ","End":"08:59.990","Text":"Here, I multiply by x and here I multiply by x,"},{"Start":"08:59.990 ","End":"09:05.300","Text":"but it will cancel so it\u0027s just like multiplying by 1 in the numerator."},{"Start":"09:05.300 ","End":"09:08.030","Text":"The other thing I have to mention is when you have a fraction,"},{"Start":"09:08.030 ","End":"09:09.830","Text":"the x cannot be 0."},{"Start":"09:09.830 ","End":"09:12.980","Text":"If we somehow get an answer of x=0,"},{"Start":"09:12.980 ","End":"09:17.285","Text":"we have to disqualify it because we can\u0027t have 0 in the denominator."},{"Start":"09:17.285 ","End":"09:21.040","Text":"Just make sure to remember at the end to see that x is not 0."},{"Start":"09:21.040 ","End":"09:27.470","Text":"Continuing over here, we get from here minus x times"},{"Start":"09:27.470 ","End":"09:35.985","Text":"2x minus 4 and here just minus 16x minus 3."},{"Start":"09:35.985 ","End":"09:43.160","Text":"We have minus 2x^2 from this product and then plus 4x from"},{"Start":"09:43.160 ","End":"09:51.410","Text":"this times this equals minus 16x and plus 16 times 3 is 48."},{"Start":"09:51.410 ","End":"09:53.420","Text":"Bring it all to the left,"},{"Start":"09:53.420 ","End":"10:03.425","Text":"minus 2x^2 plus 4x and then plus 16x minus 48=0."},{"Start":"10:03.425 ","End":"10:05.760","Text":"Combine these two,"},{"Start":"10:05.760 ","End":"10:12.020","Text":"minus 2x^2 plus 20x minus 48=0."},{"Start":"10:12.020 ","End":"10:16.490","Text":"These things all are divisible by 2, so let\u0027s divide by 2."},{"Start":"10:16.490 ","End":"10:17.975","Text":"While we\u0027re at it,"},{"Start":"10:17.975 ","End":"10:20.700","Text":"I usually like my a to be positive."},{"Start":"10:20.700 ","End":"10:22.645","Text":"Let\u0027s divide by minus 2."},{"Start":"10:22.645 ","End":"10:28.640","Text":"Here we have x^2 here dividing by minus 2 means I\u0027ve got changed sine tan x,"},{"Start":"10:28.640 ","End":"10:29.780","Text":"thereby minus 2,"},{"Start":"10:29.780 ","End":"10:33.395","Text":"I get plus 24=0."},{"Start":"10:33.395 ","End":"10:35.930","Text":"Time for the quadratic formula,"},{"Start":"10:35.930 ","End":"10:38.450","Text":"x is equal to minus b,"},{"Start":"10:38.450 ","End":"10:46.155","Text":"which is 10 plus or minus the square root of b^2 minus 4ac,"},{"Start":"10:46.155 ","End":"10:49.200","Text":"4 times 1 times 24,"},{"Start":"10:49.200 ","End":"10:50.880","Text":"all over 2a,"},{"Start":"10:50.880 ","End":"10:53.010","Text":"which is 2 times 1."},{"Start":"10:53.010 ","End":"10:57.710","Text":"Let\u0027s see, we get 10 plus or minus,"},{"Start":"10:57.710 ","End":"11:00.950","Text":"and I\u0027ll figure this out in a second over 2."},{"Start":"11:00.950 ","End":"11:07.405","Text":"Well, let\u0027s see, we can figure this out 4 times 1 times 24 is just 4 times 24 is 96."},{"Start":"11:07.405 ","End":"11:11.970","Text":"This is 100, 100 minus 96 is 4."},{"Start":"11:11.970 ","End":"11:14.315","Text":"We have plus or minus the square root of 4,"},{"Start":"11:14.315 ","End":"11:16.536","Text":"but we know the square root of 4 is 2,"},{"Start":"11:16.536 ","End":"11:19.650","Text":"it\u0027s 10 plus 2 over 2,"},{"Start":"11:19.650 ","End":"11:22.590","Text":"which is 12 over 2 is 6,"},{"Start":"11:22.590 ","End":"11:27.420","Text":"and 10 minus 2/2 is 8/2 is 4."},{"Start":"11:27.420 ","End":"11:32.295","Text":"My two answers for x, x=6 or x=4."},{"Start":"11:32.295 ","End":"11:33.770","Text":"If we had more time,"},{"Start":"11:33.770 ","End":"11:39.150","Text":"I would substitute them in here but we\u0027ll leave it at that."}],"ID":8123},{"Watched":false,"Name":"Exercise 9","Duration":"16m 11s","ChapterTopicVideoID":8031,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8031.jpeg","UploadDate":"2020-09-30T14:47:41.5430000","DurationForVideoObject":"PT16M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this exercise, we have a pair of exponential equations to solve,"},{"Start":"00:04.440 ","End":"00:07.800","Text":"let\u0027s begin with the first one and note that I have"},{"Start":"00:07.800 ","End":"00:11.489","Text":"a formula sheet provided for exponents."},{"Start":"00:11.489 ","End":"00:14.670","Text":"The idea is to get the equation to be"},{"Start":"00:14.670 ","End":"00:18.730","Text":"the same base with different exponents and then we compare the exponents,"},{"Start":"00:18.730 ","End":"00:20.655","Text":"so what would be that base?"},{"Start":"00:20.655 ","End":"00:23.640","Text":"Certainly we could take 2 as a base because"},{"Start":"00:23.640 ","End":"00:27.420","Text":"everything significant here appears to be a power of 2,"},{"Start":"00:27.420 ","End":"00:29.910","Text":"but be more efficient if we took 4,"},{"Start":"00:29.910 ","End":"00:31.920","Text":"because everything is also a power of 4,"},{"Start":"00:31.920 ","End":"00:34.590","Text":"16 is 4^2 and 64 is 4^3,"},{"Start":"00:34.590 ","End":"00:36.120","Text":"so let\u0027s go for 4."},{"Start":"00:36.120 ","End":"00:38.729","Text":"So on the left-hand side,"},{"Start":"00:38.729 ","End":"00:46.050","Text":"we have the xth root of 4^x^2 plus x,"},{"Start":"00:46.050 ","End":"00:48.690","Text":"I\u0027ll leave that as is,16,"},{"Start":"00:48.690 ","End":"00:53.025","Text":"I\u0027ll write as (4^2)^x here,"},{"Start":"00:53.025 ","End":"00:56.145","Text":"and 64 is 4^3."},{"Start":"00:56.145 ","End":"01:00.260","Text":"Now let\u0027s see if we can use some of the rules of exponents."},{"Start":"01:00.260 ","End":"01:04.760","Text":"I see here the xth root and I look at this formula,"},{"Start":"01:04.760 ","End":"01:06.260","Text":"and this is what I need,"},{"Start":"01:06.260 ","End":"01:08.390","Text":"but instead of n, I want x."},{"Start":"01:08.390 ","End":"01:11.585","Text":"What we have on the left is"},{"Start":"01:11.585 ","End":"01:20.480","Text":"(4^x^2 plus x)1/x,"},{"Start":"01:20.480 ","End":"01:22.105","Text":"just using this formula."},{"Start":"01:22.105 ","End":"01:23.865","Text":"On the right-hand side,"},{"Start":"01:23.865 ","End":"01:26.670","Text":"I can use 2 formulas,"},{"Start":"01:26.670 ","End":"01:29.210","Text":"but tell you what, let\u0027s just do them 1 at a time."},{"Start":"01:29.210 ","End":"01:33.515","Text":"The first is the numerator where I\u0027m going to use this formula."},{"Start":"01:33.515 ","End":"01:35.900","Text":"This is the one we use most frequently,"},{"Start":"01:35.900 ","End":"01:40.335","Text":"an exponent of an exponent and there we multiply the exponents."},{"Start":"01:40.335 ","End":"01:42.000","Text":"On the numerator,"},{"Start":"01:42.000 ","End":"01:48.165","Text":"I multiply the 2 by the x there and get 4^2x,"},{"Start":"01:48.165 ","End":"01:49.770","Text":"and on the denominator,"},{"Start":"01:49.770 ","End":"01:57.170","Text":"4^3, and now I use this formula again on the left-hand side,"},{"Start":"01:57.170 ","End":"01:59.345","Text":"and I multiply the 2 exponents,"},{"Start":"01:59.345 ","End":"02:03.150","Text":"so I have 4 to the power of this times this,"},{"Start":"02:03.150 ","End":"02:11.535","Text":"I\u0027d like to put the 1 over x first and then times the x^2 plus x and this is equal to,"},{"Start":"02:11.535 ","End":"02:13.290","Text":"this time we\u0027re going to use"},{"Start":"02:13.290 ","End":"02:18.720","Text":"this formula and we\u0027re also going to use it from right to left on"},{"Start":"02:18.720 ","End":"02:26.925","Text":"the right-hand side where m is 2x and n is 3 and we get 4^2x minus 3."},{"Start":"02:26.925 ","End":"02:30.635","Text":"Division of exponents, we subtract the exponents."},{"Start":"02:30.635 ","End":"02:34.250","Text":"Now we\u0027ve come to the middle stage that we were hoping"},{"Start":"02:34.250 ","End":"02:38.555","Text":"for is that we have the same base on both sides."},{"Start":"02:38.555 ","End":"02:42.589","Text":"We have 4 to the power of something equals 4 to the power of something."},{"Start":"02:42.589 ","End":"02:47.510","Text":"That means that we didn\u0027t basically just throw out the bases,"},{"Start":"02:47.510 ","End":"02:49.475","Text":"we compare the exponents,"},{"Start":"02:49.475 ","End":"02:55.620","Text":"and so we get that 1/x(x^2+x)=2x-3."},{"Start":"02:59.260 ","End":"03:03.665","Text":"Now this is an equation with fractions denominators"},{"Start":"03:03.665 ","End":"03:09.215","Text":"and there we have to be careful that we don\u0027t have a denominator of 0."},{"Start":"03:09.215 ","End":"03:14.905","Text":"We must remember at the end to check that our x does not come out 0,"},{"Start":"03:14.905 ","End":"03:17.420","Text":"maybe I\u0027ll make a little note of it here,"},{"Start":"03:17.420 ","End":"03:20.030","Text":"x not equal to 0,"},{"Start":"03:20.030 ","End":"03:22.160","Text":"I have to check at the end to make sure I didn\u0027t"},{"Start":"03:22.160 ","End":"03:24.475","Text":"forget about that because 0 is not allowed."},{"Start":"03:24.475 ","End":"03:28.730","Text":"Easiest thing to do here is to multiply both sides by x,"},{"Start":"03:28.730 ","End":"03:30.575","Text":"the x with the x will go,"},{"Start":"03:30.575 ","End":"03:32.760","Text":"so I will get"},{"Start":"03:37.390 ","End":"03:42.244","Text":"x^2+x=x(2x-3) and that will give us a quadratic equation,"},{"Start":"03:42.244 ","End":"03:43.880","Text":"which I\u0027m doing over here,"},{"Start":"03:43.880 ","End":"03:50.660","Text":"we get that x^2+x=x"},{"Start":"03:50.660 ","End":"03:55.755","Text":"time 2x is 2x^2-x times 3 is 3x,"},{"Start":"03:55.755 ","End":"04:00.620","Text":"let\u0027s put everything on the left in order."},{"Start":"04:00.620 ","End":"04:04.830","Text":"First the x^2s, we have x^2 from here minus 2x^2,"},{"Start":"04:04.830 ","End":"04:07.005","Text":"so that\u0027s minus x^2."},{"Start":"04:07.005 ","End":"04:12.435","Text":"Here we have plus x and then again another plus 3x,"},{"Start":"04:12.435 ","End":"04:14.985","Text":"so altogether, plus 4x,"},{"Start":"04:14.985 ","End":"04:19.065","Text":"just know just numbers equals 0,"},{"Start":"04:19.065 ","End":"04:27.230","Text":"and this is 1 of those equations with missing c. What we do is we take x"},{"Start":"04:27.230 ","End":"04:37.130","Text":"outside the brackets and we get x(-x+4)=0."},{"Start":"04:37.130 ","End":"04:39.754","Text":"Remember, when a product is 0,"},{"Start":"04:39.754 ","End":"04:43.160","Text":"the only way that can happen is if 1 or the other is 0."},{"Start":"04:43.160 ","End":"04:45.890","Text":"We do get that"},{"Start":"04:45.890 ","End":"04:54.425","Text":"either x=0 or -x+4=0."},{"Start":"04:54.425 ","End":"04:59.970","Text":"This one clearly gives us that x=4 and I was"},{"Start":"04:59.970 ","End":"05:02.870","Text":"about to highlight both solutions but"},{"Start":"05:02.870 ","End":"05:06.013","Text":"I\u0027m only highlighting this one because I remembered,"},{"Start":"05:06.013 ","End":"05:07.268","Text":"and here it is I wrote it down,"},{"Start":"05:07.268 ","End":"05:08.915","Text":"that x cannot be 0,"},{"Start":"05:08.915 ","End":"05:12.985","Text":"so the only solution is x=4."},{"Start":"05:12.985 ","End":"05:16.250","Text":"This one is not allowed and it isn\u0027t"},{"Start":"05:16.250 ","End":"05:19.220","Text":"even allowed even from the original equation because there is"},{"Start":"05:19.220 ","End":"05:26.000","Text":"no such thing as a 0th root, just isn\u0027t defined."},{"Start":"05:26.000 ","End":"05:29.900","Text":"Let\u0027s check that this really is a solution,"},{"Start":"05:29.900 ","End":"05:31.940","Text":"I\u0027m suddenly having doubts,"},{"Start":"05:31.940 ","End":"05:35.000","Text":"so let\u0027s go back to the original equation."},{"Start":"05:35.000 ","End":"05:38.660","Text":"I\u0027ll call the left-hand side, LHS,"},{"Start":"05:38.660 ","End":"05:41.270","Text":"that\u0027s customary abbreviation and this will be"},{"Start":"05:41.270 ","End":"05:46.310","Text":"the right-hand side and I\u0027ll check each 1 separately and see that we get equal things."},{"Start":"05:46.310 ","End":"05:52.305","Text":"The left-hand side is the 4th root,"},{"Start":"05:52.305 ","End":"05:59.295","Text":"because x is (4^4)^2 plus 4,"},{"Start":"05:59.295 ","End":"06:05.220","Text":"this will equal the 4th root of 4 to the power of,"},{"Start":"06:05.220 ","End":"06:08.850","Text":"let\u0027s see, 16 plus 4 is 20."},{"Start":"06:08.850 ","End":"06:14.120","Text":"Now the 4th root means take to the power of 1/4 from this formula,"},{"Start":"06:14.120 ","End":"06:20.513","Text":"so it\u0027s (4^20)^1/4,"},{"Start":"06:20.513 ","End":"06:25.275","Text":"which is equal to 4^20 times 1/4,"},{"Start":"06:25.275 ","End":"06:33.150","Text":"20 times 1/4 is 5 and 4^5 happen to know is 1,024."},{"Start":"06:33.150 ","End":"06:34.800","Text":"That\u0027s the left-hand side,"},{"Start":"06:34.800 ","End":"06:37.835","Text":"let\u0027s see what the right-hand side comes out to."},{"Start":"06:37.835 ","End":"06:41.135","Text":"Here we have 16^x,"},{"Start":"06:41.135 ","End":"06:50.790","Text":"16^4 over 64 and what we can do is,"},{"Start":"06:50.790 ","End":"06:54.390","Text":"I could use the calculator,"},{"Start":"06:54.390 ","End":"06:56.009","Text":"I don\u0027t want to use a calculator,"},{"Start":"06:56.009 ","End":"06:59.869","Text":"so what I could do is that 16 is 4^2,"},{"Start":"06:59.869 ","End":"07:02.546","Text":"just like we did in the exercise,"},{"Start":"07:02.546 ","End":"07:09.070","Text":"we can say this is (4^2)^4 and 64 we could write as 4^3,"},{"Start":"07:09.070 ","End":"07:17.800","Text":"and then we\u0027ll get 4^8 over 4^3."},{"Start":"07:17.800 ","End":"07:25.180","Text":"We can use this formula again and say that this is equal to 4^5."},{"Start":"07:25.180 ","End":"07:27.400","Text":"I guess I didn\u0027t need to continue to the end,"},{"Start":"07:27.400 ","End":"07:28.705","Text":"but we might as well."},{"Start":"07:28.705 ","End":"07:30.985","Text":"This is 1,024."},{"Start":"07:30.985 ","End":"07:34.419","Text":"Just happen to know this because I\u0027ve encountered it before,"},{"Start":"07:34.419 ","End":"07:36.475","Text":"so I didn\u0027t need a calculator."},{"Start":"07:36.475 ","End":"07:39.235","Text":"This is verified as a solution."},{"Start":"07:39.235 ","End":"07:43.210","Text":"Now let\u0027s go on to part b."},{"Start":"07:43.210 ","End":"07:45.280","Text":"If we need the formulas,"},{"Start":"07:45.280 ","End":"07:46.690","Text":"I\u0027ll bring them down."},{"Start":"07:46.690 ","End":"07:48.819","Text":"I\u0027d better go and get those formulas."},{"Start":"07:48.819 ","End":"07:52.930","Text":"There we are. I already see we have all these roots."},{"Start":"07:52.930 ","End":"07:58.090","Text":"It looks like we\u0027re going to need this formula. Let\u0027s begin."},{"Start":"07:58.090 ","End":"08:00.895","Text":"We need to decide on what base we\u0027re going for."},{"Start":"08:00.895 ","End":"08:02.496","Text":"Looks like 3 is it,"},{"Start":"08:02.496 ","End":"08:06.415","Text":"because 81 is 3^4. We know that."},{"Start":"08:06.415 ","End":"08:09.820","Text":"Notice that 243 is just 3 times 81."},{"Start":"08:09.820 ","End":"08:12.280","Text":"This will be 3^5."},{"Start":"08:12.280 ","End":"08:17.950","Text":"What I have here is 3^4, that\u0027s the 81."},{"Start":"08:17.950 ","End":"08:20.110","Text":"Now, I\u0027m going to do 2 steps in 1."},{"Start":"08:20.110 ","End":"08:27.190","Text":"The xth root would be to the power of 1 over x. I have an xth root."},{"Start":"08:27.190 ","End":"08:31.660","Text":"I\u0027ll put this in brackets and again put 1 over x here."},{"Start":"08:31.660 ","End":"08:33.100","Text":"That\u0027s the left-hand side."},{"Start":"08:33.100 ","End":"08:35.350","Text":"Now here, remember I said this is 3^5."},{"Start":"08:35.350 ","End":"08:41.890","Text":"You can check 3 times 3 times 3 times 3 times 3 is 243, that\u0027s 3^5."},{"Start":"08:41.890 ","End":"08:45.820","Text":"Now I\u0027m going to use another formula here."},{"Start":"08:45.820 ","End":"08:48.085","Text":"I mean this one."},{"Start":"08:48.085 ","End":"08:49.825","Text":"I\u0027m looking at this as a fraction."},{"Start":"08:49.825 ","End":"08:50.980","Text":"There was another way to go."},{"Start":"08:50.980 ","End":"08:55.150","Text":"I could\u0027ve used the one over and I usually use this one,"},{"Start":"08:55.150 ","End":"08:57.580","Text":"but just for a change into practice."},{"Start":"08:57.580 ","End":"08:59.890","Text":"This law will do it the other way."},{"Start":"08:59.890 ","End":"09:01.525","Text":"We\u0027ll get the same answer of course."},{"Start":"09:01.525 ","End":"09:10.825","Text":"We get 1^x over 3^x."},{"Start":"09:10.825 ","End":"09:15.995","Text":"This x and this x are the n in this formula."},{"Start":"09:15.995 ","End":"09:21.450","Text":"Now here, we can use this formula twice in a row."},{"Start":"09:21.450 ","End":"09:25.085","Text":"I\u0027ve got something to the power of something and then to the power of something."},{"Start":"09:25.085 ","End":"09:29.080","Text":"We can multiply them all together instead of doing it two at a time."},{"Start":"09:29.080 ","End":"09:33.925","Text":"It\u0027s 3^4. Then we\u0027d get 1 over x we\u0027d multiply it by 1 over x."},{"Start":"09:33.925 ","End":"09:36.235","Text":"Then we\u0027d multiply by 1 over x again."},{"Start":"09:36.235 ","End":"09:38.635","Text":"Let\u0027s do it all in 1 stroke."},{"Start":"09:38.635 ","End":"09:41.800","Text":"Here we have 3^5."},{"Start":"09:41.800 ","End":"09:43.990","Text":"We\u0027re going to put it in with the numerator."},{"Start":"09:43.990 ","End":"09:46.480","Text":"1 to the power of anything is 1."},{"Start":"09:46.480 ","End":"09:49.000","Text":"This just, ignore that."},{"Start":"09:49.000 ","End":"09:50.395","Text":"1 times 1 times,"},{"Start":"09:50.395 ","End":"09:51.730","Text":"doesn\u0027t matter is 1."},{"Start":"09:51.730 ","End":"09:54.700","Text":"On the denominator we\u0027re using this rule."},{"Start":"09:54.700 ","End":"09:59.755","Text":"It\u0027s 3^x times x."},{"Start":"09:59.755 ","End":"10:02.320","Text":"Let\u0027s see what we have,"},{"Start":"10:02.320 ","End":"10:06.490","Text":"and why does this 5 look like a 6? That\u0027s better."},{"Start":"10:06.490 ","End":"10:12.580","Text":"On the left-hand side we have 3^4 times 1 over x times 1"},{"Start":"10:12.580 ","End":"10:18.535","Text":"over x is 4 over x^2."},{"Start":"10:18.535 ","End":"10:22.825","Text":"Let\u0027s do 2 steps in 1."},{"Start":"10:22.825 ","End":"10:26.095","Text":"The denominator is 3^(x^2)."},{"Start":"10:26.095 ","End":"10:29.830","Text":"I mean, I always mentally see the x^2 there."},{"Start":"10:29.830 ","End":"10:32.200","Text":"I\u0027m going to use this formula."},{"Start":"10:32.200 ","End":"10:36.085","Text":"This one from right to left. We can subtract."},{"Start":"10:36.085 ","End":"10:43.255","Text":"All in all we can say that the right-hand side is 3^5. This is over 3 to the x^2."},{"Start":"10:43.255 ","End":"10:47.050","Text":"It\u0027s 5 minus x^2."},{"Start":"10:47.050 ","End":"10:51.400","Text":"Now, we\u0027re at the point where we have the same base."},{"Start":"10:51.400 ","End":"10:54.680","Text":"I mean the base 3 of course."},{"Start":"10:54.750 ","End":"10:58.150","Text":"The way it works is that we can now ignore"},{"Start":"10:58.150 ","End":"11:02.320","Text":"the bases or throw them out and compare the exponents."},{"Start":"11:02.320 ","End":"11:07.210","Text":"We get the equation,4 over x^2"},{"Start":"11:07.210 ","End":"11:13.180","Text":"equals 5 minus x^2."},{"Start":"11:13.180 ","End":"11:17.080","Text":"Note here that x cannot be 0,"},{"Start":"11:17.080 ","End":"11:20.350","Text":"because then that would make the denominator 0."},{"Start":"11:20.350 ","End":"11:24.715","Text":"I\u0027m making a note here that x cannot be 0."},{"Start":"11:24.715 ","End":"11:26.380","Text":"It\u0027s already couldn\u0027t be 0 here."},{"Start":"11:26.380 ","End":"11:29.740","Text":"In fact, right from the start I could\u0027ve said that x is not 0,"},{"Start":"11:29.740 ","End":"11:32.870","Text":"because we can\u0027t take 0th root."},{"Start":"11:33.060 ","End":"11:38.650","Text":"Let\u0027s multiply both sides by x^2."},{"Start":"11:38.650 ","End":"11:39.700","Text":"Make a note to that."},{"Start":"11:39.700 ","End":"11:43.060","Text":"We\u0027re multiplying both sides of the equation."},{"Start":"11:43.060 ","End":"11:46.360","Text":"On the left it cancels and we just have 4."},{"Start":"11:46.360 ","End":"11:49.915","Text":"Here we have, I\u0027ll put the x^2 in front."},{"Start":"11:49.915 ","End":"11:55.180","Text":"x^2 times 5 minus x^2."},{"Start":"11:55.180 ","End":"12:02.845","Text":"Let\u0027s see, 4 equals opening brackets, 5x^2 minus x^4."},{"Start":"12:02.845 ","End":"12:05.620","Text":"Put everything on the left-hand side."},{"Start":"12:05.620 ","End":"12:15.220","Text":"We have plus x^4 minus 5x^2 plus 4 equals 0."},{"Start":"12:15.220 ","End":"12:20.815","Text":"The equation that we got now is something called a bi-quadratic equation."},{"Start":"12:20.815 ","End":"12:24.875","Text":"I don\u0027t know if you have studied this or not."},{"Start":"12:24.875 ","End":"12:30.825","Text":"If not, you can either skip the rest of the lesson or you can stay and learn something."},{"Start":"12:30.825 ","End":"12:33.185","Text":"I\u0027ll continue solving it anyway."},{"Start":"12:33.185 ","End":"12:36.940","Text":"It\u0027s bi-quadratic, meaning it\u0027s a power of 4 equation."},{"Start":"12:36.940 ","End":"12:38.440","Text":"But the odd powers are missing."},{"Start":"12:38.440 ","End":"12:40.570","Text":"There\u0027s no x^3 and there is no x."},{"Start":"12:40.570 ","End":"12:42.918","Text":"There\u0027s a standard way of solving this,"},{"Start":"12:42.918 ","End":"12:45.100","Text":"and that is to substitute."},{"Start":"12:45.100 ","End":"12:47.755","Text":"Instead of x^2, we\u0027re going to put y."},{"Start":"12:47.755 ","End":"12:49.630","Text":"If y is x^2,"},{"Start":"12:49.630 ","End":"12:54.985","Text":"then y^2 becomes (x^2)^2 which is x^4."},{"Start":"12:54.985 ","End":"13:05.335","Text":"What we get here is y^2 minus 5y^2 is y plus 4 equals 0."},{"Start":"13:05.335 ","End":"13:07.600","Text":"Now, it\u0027s a regular quadratic equation."},{"Start":"13:07.600 ","End":"13:09.655","Text":"But in y not in x."},{"Start":"13:09.655 ","End":"13:12.505","Text":"We solve it using the formula,"},{"Start":"13:12.505 ","End":"13:18.564","Text":"y equals minus b is plus 5 plus or minus"},{"Start":"13:18.564 ","End":"13:27.565","Text":"the square root of b^2 is 5^2 minus 4. a is 1, c is 4."},{"Start":"13:27.565 ","End":"13:29.545","Text":"All this over 2a."},{"Start":"13:29.545 ","End":"13:33.760","Text":"Which is 2 times 1 which is 2. Let\u0027s see."},{"Start":"13:33.760 ","End":"13:35.200","Text":"Under the square root sign,"},{"Start":"13:35.200 ","End":"13:42.250","Text":"we have 25 minus 16 is 9."},{"Start":"13:42.250 ","End":"13:46.180","Text":"It\u0027s 5 plus or minus well,"},{"Start":"13:46.180 ","End":"13:49.090","Text":"the square root of 9, which is 3."},{"Start":"13:49.090 ","End":"13:53.860","Text":"It\u0027s plus or minus 3 all over 2."},{"Start":"13:53.860 ","End":"13:57.730","Text":"Then we branch off according to plus or minus."},{"Start":"13:57.730 ","End":"14:01.525","Text":"If it\u0027s plus, 5 plus 3 over 2a over 2 is 4."},{"Start":"14:01.525 ","End":"14:03.310","Text":"If it\u0027s minus, 5 minus 3 is 2,"},{"Start":"14:03.310 ","End":"14:04.600","Text":"over 2 is 1."},{"Start":"14:04.600 ","End":"14:06.925","Text":"We have 2 solutions for y."},{"Start":"14:06.925 ","End":"14:09.160","Text":"But that\u0027s not the end of it,"},{"Start":"14:09.160 ","End":"14:13.525","Text":"because y is x^2."},{"Start":"14:13.525 ","End":"14:16.895","Text":"This gives us two more equations."},{"Start":"14:16.895 ","End":"14:20.940","Text":"This gives us that x^2=4."},{"Start":"14:21.010 ","End":"14:27.430","Text":"This one gives us that x^2 is 1,"},{"Start":"14:27.430 ","End":"14:29.485","Text":"because these are the values of y."},{"Start":"14:29.485 ","End":"14:31.660","Text":"But y is x^2."},{"Start":"14:31.660 ","End":"14:37.475","Text":"This one we solved just by taking the square root plus or minus."},{"Start":"14:37.475 ","End":"14:42.215","Text":"We have x is plus or minus the square root of 4."},{"Start":"14:42.215 ","End":"14:44.630","Text":"In other words, plus or minus 2."},{"Start":"14:44.630 ","End":"14:50.765","Text":"This gives us that x equals plus or minus square root of 1 which is 1."},{"Start":"14:50.765 ","End":"14:55.670","Text":"Now, at first sight it appears that there are 4 solutions for x."},{"Start":"14:55.670 ","End":"15:01.265","Text":"It appears that we have the possibility that x is either 2,"},{"Start":"15:01.265 ","End":"15:05.700","Text":"minus 2, 1, or minus 1."},{"Start":"15:05.700 ","End":"15:08.225","Text":"However, there are some restrictions."},{"Start":"15:08.225 ","End":"15:11.720","Text":"First of all, we have to check that x is not equal to 0."},{"Start":"15:11.720 ","End":"15:13.340","Text":"That\u0027s what I mentioned before,"},{"Start":"15:13.340 ","End":"15:16.070","Text":"because x appears in the denominator here."},{"Start":"15:16.070 ","End":"15:18.320","Text":"Well, certainly none of these are 0,"},{"Start":"15:18.320 ","End":"15:20.944","Text":"so far we still have 4 solutions."},{"Start":"15:20.944 ","End":"15:25.220","Text":"However, there is another restriction which I don\u0027t say each time,"},{"Start":"15:25.220 ","End":"15:30.070","Text":"but whenever you see the xth root of something or the nth root."},{"Start":"15:30.070 ","End":"15:31.890","Text":"In fact, in this formula,"},{"Start":"15:31.890 ","End":"15:37.625","Text":"one should really add each time that n has to be a positive whole number 1,"},{"Start":"15:37.625 ","End":"15:40.355","Text":"2, 3, 4 etc."},{"Start":"15:40.355 ","End":"15:44.810","Text":"We can\u0027t take a negative root or a fractional root."},{"Start":"15:44.810 ","End":"15:47.210","Text":"Here in the original question,"},{"Start":"15:47.210 ","End":"15:50.480","Text":"we have the xth root once inside the other,"},{"Start":"15:50.480 ","End":"15:53.825","Text":"but that\u0027s certainly implies that x is"},{"Start":"15:53.825 ","End":"15:58.130","Text":"a positive integer that rules out the negative ones."},{"Start":"15:58.130 ","End":"16:00.095","Text":"This is ruled out,"},{"Start":"16:00.095 ","End":"16:01.520","Text":"and this is ruled out."},{"Start":"16:01.520 ","End":"16:12.030","Text":"Really, I have to say that the only 2 solutions are x=2 and x=1, and we are done."}],"ID":8124},{"Watched":false,"Name":"Exercise 10","Duration":"11m 19s","ChapterTopicVideoID":8032,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8032.jpeg","UploadDate":"2020-09-30T14:48:47.9730000","DurationForVideoObject":"PT11M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"In this exercise, we have to solve a couple of exponential equations."},{"Start":"00:05.175 ","End":"00:08.295","Text":"We\u0027ll start with the first one."},{"Start":"00:08.295 ","End":"00:15.525","Text":"I already provided a table of formulas for exponents. Let\u0027s begin."},{"Start":"00:15.525 ","End":"00:21.155","Text":"What we have to do is find what base we want to express both sides in terms of."},{"Start":"00:21.155 ","End":"00:23.460","Text":"It looks like if see 4,"},{"Start":"00:23.460 ","End":"00:25.005","Text":"32, and 8,"},{"Start":"00:25.005 ","End":"00:27.330","Text":"they\u0027re all powers of 2."},{"Start":"00:27.330 ","End":"00:30.660","Text":"Let\u0027s write everything in terms of 2."},{"Start":"00:30.660 ","End":"00:36.260","Text":"Now, notice also that before I even start with the 2,"},{"Start":"00:36.260 ","End":"00:40.820","Text":"I\u0027d like to write this thing using this formula."},{"Start":"00:40.820 ","End":"00:42.645","Text":"I\u0027ve used this before,"},{"Start":"00:42.645 ","End":"00:50.345","Text":"and the views did also twice in the same time because 4^x^x would use the formula,"},{"Start":"00:50.345 ","End":"00:52.490","Text":"and then, again, to the power of x,"},{"Start":"00:52.490 ","End":"00:56.630","Text":"we can just say that if we have a chain of exponents,"},{"Start":"00:56.630 ","End":"00:58.310","Text":"we just have to multiply them all."},{"Start":"00:58.310 ","End":"01:00.155","Text":"It could even be 3 or more."},{"Start":"01:00.155 ","End":"01:02.165","Text":"It doesn\u0027t have to just be 2."},{"Start":"01:02.165 ","End":"01:04.250","Text":"Things are simplified quite a bit."},{"Start":"01:04.250 ","End":"01:08.720","Text":"If I write it as 4^x times x times x,"},{"Start":"01:08.720 ","End":"01:10.220","Text":"of course this is x^3."},{"Start":"01:10.220 ","End":"01:11.480","Text":"I\u0027ll just leave it for the moment,"},{"Start":"01:11.480 ","End":"01:21.165","Text":"so you can see times 32 and I\u0027ll write that as (2^5)^x^2."},{"Start":"01:21.165 ","End":"01:22.365","Text":"Here the 8,"},{"Start":"01:22.365 ","End":"01:25.750","Text":"I\u0027ll write as (2^3)^x."},{"Start":"01:25.910 ","End":"01:28.545","Text":"Let\u0027s see what we have now."},{"Start":"01:28.545 ","End":"01:31.320","Text":"This is (4^x)^3,"},{"Start":"01:31.320 ","End":"01:33.575","Text":"4 is 2^2,"},{"Start":"01:33.575 ","End":"01:36.560","Text":"and then this is to the power of x^3."},{"Start":"01:36.560 ","End":"01:40.145","Text":"Then we have times 2 to the power of,"},{"Start":"01:40.145 ","End":"01:41.749","Text":"using this formula, again,"},{"Start":"01:41.749 ","End":"01:44.545","Text":"it\u0027s 5 times x^2."},{"Start":"01:44.545 ","End":"01:49.210","Text":"Again with this formula, this is 2^3x."},{"Start":"01:49.210 ","End":"02:00.110","Text":"Let me just rewrite this as this thing here is (2^2)x^3."},{"Start":"02:00.110 ","End":"02:04.265","Text":"I\u0027m just saving myself a step or row of writing."},{"Start":"02:04.265 ","End":"02:10.535","Text":"Now I can use the formula for this one for the product."},{"Start":"02:10.535 ","End":"02:12.470","Text":"When I have a product of exponents,"},{"Start":"02:12.470 ","End":"02:14.435","Text":"I add the exponents."},{"Start":"02:14.435 ","End":"02:19.685","Text":"Here I get (2)^2x^3,"},{"Start":"02:19.685 ","End":"02:21.395","Text":"which we wrote over here."},{"Start":"02:21.395 ","End":"02:24.949","Text":"This is 5x^2."},{"Start":"02:24.949 ","End":"02:28.235","Text":"I add the exponents just like in this formula,"},{"Start":"02:28.235 ","End":"02:29.420","Text":"and on the right-hand side,"},{"Start":"02:29.420 ","End":"02:30.995","Text":"it\u0027s just the same."},{"Start":"02:30.995 ","End":"02:36.740","Text":"Now we come to the desired situation of same base on both sides,"},{"Start":"02:36.740 ","End":"02:39.835","Text":"base 2 and base 2."},{"Start":"02:39.835 ","End":"02:44.105","Text":"At this point, I just throw out the bases metaphorically,"},{"Start":"02:44.105 ","End":"02:52.020","Text":"and we get 2x^3+5x^2=3x."},{"Start":"02:52.020 ","End":"02:55.645","Text":"Bringing this to the left-hand side, you know what?"},{"Start":"02:55.645 ","End":"02:57.320","Text":"I\u0027ll just, again, save a row."},{"Start":"02:57.320 ","End":"03:00.305","Text":"If I just write minus this equals 0."},{"Start":"03:00.305 ","End":"03:02.420","Text":"There changed equal to a minus,"},{"Start":"03:02.420 ","End":"03:04.510","Text":"and then I put equals 0."},{"Start":"03:04.510 ","End":"03:08.180","Text":"Now you\u0027re thinking, cubic equation power of 3."},{"Start":"03:08.180 ","End":"03:09.890","Text":"We haven\u0027t learned that."},{"Start":"03:09.890 ","End":"03:11.900","Text":"Maybe you have, maybe you haven\u0027t."},{"Start":"03:11.900 ","End":"03:16.610","Text":"But in any event, we can take x outside the brackets and then"},{"Start":"03:16.610 ","End":"03:18.950","Text":"we will be left with a quadratic equation"},{"Start":"03:18.950 ","End":"03:21.650","Text":"because there\u0027s no constant term so we could do that."},{"Start":"03:21.650 ","End":"03:29.240","Text":"It\u0027s x times 2x^2+5x-3=0."},{"Start":"03:29.240 ","End":"03:32.300","Text":"Now, we have 2 possibilities."},{"Start":"03:32.300 ","End":"03:33.830","Text":"We have a product which is 0."},{"Start":"03:33.830 ","End":"03:35.990","Text":"We have that either x is"},{"Start":"03:35.990 ","End":"03:44.900","Text":"0 or other possibility is that 2x^2 plus 5x minus 3=0."},{"Start":"03:44.900 ","End":"03:48.350","Text":"This straightaway is one of the answers."},{"Start":"03:48.350 ","End":"03:50.080","Text":"Here we have to work a bit more,"},{"Start":"03:50.080 ","End":"03:51.650","Text":"and we have to solve a quadratic,"},{"Start":"03:51.650 ","End":"03:54.230","Text":"we might get two more answers. We might not."},{"Start":"03:54.230 ","End":"03:57.440","Text":"Quadratics don\u0027t always have two solutions, but let\u0027s see."},{"Start":"03:57.440 ","End":"04:04.100","Text":"Over here we are going to use the quadratic formula where this is my a, b,"},{"Start":"04:04.100 ","End":"04:10.520","Text":"and c. X is equal to minus b plus or"},{"Start":"04:10.520 ","End":"04:18.005","Text":"minus the square root of b^2 minus 4a,"},{"Start":"04:18.005 ","End":"04:20.495","Text":"c is minus 3,"},{"Start":"04:20.495 ","End":"04:23.165","Text":"and all this is over 2a,"},{"Start":"04:23.165 ","End":"04:25.210","Text":"which is 2 times 2."},{"Start":"04:25.210 ","End":"04:27.395","Text":"What we get is,"},{"Start":"04:27.395 ","End":"04:29.545","Text":"let\u0027s do what\u0027s under the square root first."},{"Start":"04:29.545 ","End":"04:35.580","Text":"4 times 2 times 3 is 8 times 3 is 24."},{"Start":"04:35.580 ","End":"04:38.340","Text":"Now it\u0027s plus 24."},{"Start":"04:38.340 ","End":"04:45.325","Text":"This is 25 plus 24 is 49 over 4."},{"Start":"04:45.325 ","End":"04:49.160","Text":"Now, just make a note that the square root of 49 is 7."},{"Start":"04:49.160 ","End":"04:50.575","Text":"Everyone knows that."},{"Start":"04:50.575 ","End":"04:53.010","Text":"7 times 7 is 49."},{"Start":"04:53.010 ","End":"04:58.020","Text":"What we have is if it\u0027s minus 5 plus 7 over 4,"},{"Start":"04:58.020 ","End":"05:01.199","Text":"it\u0027s 2 over 4. It\u0027s a half."},{"Start":"05:01.199 ","End":"05:05.410","Text":"If it\u0027s minus 5 minus 7 it\u0027s minus 12."},{"Start":"05:05.410 ","End":"05:09.565","Text":"Over 4 is minus 3."},{"Start":"05:09.565 ","End":"05:15.545","Text":"This gives us a possibility that we have one possibility."},{"Start":"05:15.545 ","End":"05:18.230","Text":"Then maybe I\u0027ll collect them."},{"Start":"05:18.230 ","End":"05:19.955","Text":"I\u0027ll highlight this,"},{"Start":"05:19.955 ","End":"05:22.700","Text":"and this, and this,"},{"Start":"05:22.700 ","End":"05:32.590","Text":"and really like to collect them all and say that x=0 or 1 over 2 or minus 3,"},{"Start":"05:32.590 ","End":"05:34.850","Text":"meaning that all 3 of these are possible."},{"Start":"05:34.850 ","End":"05:38.045","Text":"Each of these should work if we substitute."},{"Start":"05:38.045 ","End":"05:39.860","Text":"Let\u0027s just check one of them."},{"Start":"05:39.860 ","End":"05:42.185","Text":"Let\u0027s check if the 0 works."},{"Start":"05:42.185 ","End":"05:48.045","Text":"I\u0027m going to go for x=0 in the original equation."},{"Start":"05:48.045 ","End":"05:56.325","Text":"What I have is 4^0^0^0"},{"Start":"05:56.325 ","End":"06:00.565","Text":"times 32^0^2."},{"Start":"06:00.565 ","End":"06:01.970","Text":"Is this equal?"},{"Start":"06:01.970 ","End":"06:05.630","Text":"That\u0027s what we have to check 8^0."},{"Start":"06:05.630 ","End":"06:07.640","Text":"Well, if you think about it,"},{"Start":"06:07.640 ","End":"06:10.730","Text":"it\u0027s true because 4^0 is 1,"},{"Start":"06:10.730 ","End":"06:13.460","Text":"and then to the power of 0 is still 1,"},{"Start":"06:13.460 ","End":"06:15.980","Text":"and to the power of 0 is still 1,"},{"Start":"06:15.980 ","End":"06:20.705","Text":"and 32^0^2 is 1,"},{"Start":"06:20.705 ","End":"06:23.055","Text":"and 8^0 is 1,"},{"Start":"06:23.055 ","End":"06:25.050","Text":"and 1 times 1=1."},{"Start":"06:25.050 ","End":"06:31.414","Text":"Yes. We verified 0 and I believe that the other two are probably correct also,"},{"Start":"06:31.414 ","End":"06:34.360","Text":"but can\u0027t waste all day checking."},{"Start":"06:34.360 ","End":"06:36.260","Text":"From time to time, you should do it."},{"Start":"06:36.260 ","End":"06:38.668","Text":"Moving on to b,"},{"Start":"06:38.668 ","End":"06:42.475","Text":"and taking the formula with me. Why not?"},{"Start":"06:42.475 ","End":"06:44.840","Text":"Let\u0027s see what we have here."},{"Start":"06:44.840 ","End":"06:50.500","Text":"Looks like here we want to go for base 3 because everything say power of 3."},{"Start":"06:50.500 ","End":"06:56.780","Text":"3^x times 1 over"},{"Start":"06:56.780 ","End":"07:03.065","Text":"27 is 3^3 to the power of square root x."},{"Start":"07:03.065 ","End":"07:08.095","Text":"This equals 9 is 3^2^5."},{"Start":"07:08.095 ","End":"07:09.740","Text":"Let\u0027s see what we get."},{"Start":"07:09.740 ","End":"07:12.635","Text":"I\u0027ll just put this in the numerator."},{"Start":"07:12.635 ","End":"07:18.900","Text":"We get 3^x,"},{"Start":"07:18.900 ","End":"07:20.870","Text":"now I\u0027m using this formula for this one,"},{"Start":"07:20.870 ","End":"07:24.320","Text":"exponent of an exponent, power of a power."},{"Start":"07:24.320 ","End":"07:29.335","Text":"Just multiply them, so I get 3 square root of x."},{"Start":"07:29.335 ","End":"07:31.220","Text":"On the right-hand side,"},{"Start":"07:31.220 ","End":"07:32.630","Text":"also using this formula,"},{"Start":"07:32.630 ","End":"07:35.300","Text":"I\u0027ve got 3^2 times 5."},{"Start":"07:35.300 ","End":"07:37.535","Text":"I\u0027ll write that straightaway as 10."},{"Start":"07:37.535 ","End":"07:41.106","Text":"Now I want to use this formula,"},{"Start":"07:41.106 ","End":"07:43.445","Text":"the one with subtraction, this one,"},{"Start":"07:43.445 ","End":"07:46.550","Text":"because we have power over a power."},{"Start":"07:46.550 ","End":"07:49.415","Text":"I\u0027m going to use this formula."},{"Start":"07:49.415 ","End":"07:54.320","Text":"This means that we have to subtract this exponent minus this exponent,"},{"Start":"07:54.320 ","End":"07:59.060","Text":"3^x minus 3 times square root of x."},{"Start":"07:59.060 ","End":"08:02.605","Text":"This equals 3^10."},{"Start":"08:02.605 ","End":"08:08.200","Text":"Now we have the situation where we\u0027ve got the same base on both sides,"},{"Start":"08:08.200 ","End":"08:11.085","Text":"3 here and 3 here."},{"Start":"08:11.085 ","End":"08:19.160","Text":"We can compare the exponents and get that x minus 3 times square root of x=10."},{"Start":"08:19.160 ","End":"08:21.785","Text":"Or bringing the 10 to the other side,"},{"Start":"08:21.785 ","End":"08:29.670","Text":"x minus 3 square root of x minus 10=0."},{"Start":"08:29.670 ","End":"08:32.555","Text":"This is one of those special equations,"},{"Start":"08:32.555 ","End":"08:35.360","Text":"which is not a quadratic equation,"},{"Start":"08:35.360 ","End":"08:39.545","Text":"but becomes a quadratic equation with a substitution."},{"Start":"08:39.545 ","End":"08:46.310","Text":"What we do is a standard trick is to let y equals the square root of x."},{"Start":"08:46.310 ","End":"08:48.800","Text":"But we have to make sure at the end that y comes up"},{"Start":"08:48.800 ","End":"08:51.730","Text":"positive because the square root is always positive."},{"Start":"08:51.730 ","End":"08:54.845","Text":"Then if we look at what y^2 is,"},{"Start":"08:54.845 ","End":"09:00.650","Text":"y^2 is just x because if you square root,"},{"Start":"09:00.650 ","End":"09:02.810","Text":"you\u0027re back to the number itself;"},{"Start":"09:02.810 ","End":"09:08.570","Text":"x and y both have to be positive for taking a square root positive or 0 that is,"},{"Start":"09:08.570 ","End":"09:11.135","Text":"and y also positive or 0."},{"Start":"09:11.135 ","End":"09:13.385","Text":"Now with this substitution,"},{"Start":"09:13.385 ","End":"09:15.545","Text":"this equation becomes,"},{"Start":"09:15.545 ","End":"09:17.120","Text":"and I\u0027ll continue over here,"},{"Start":"09:17.120 ","End":"09:19.565","Text":"x we said is y^2."},{"Start":"09:19.565 ","End":"09:26.570","Text":"Then minus 3 square root of x is y minus 10=0."},{"Start":"09:26.570 ","End":"09:30.200","Text":"Familiar territory quadratic equation just happens"},{"Start":"09:30.200 ","End":"09:34.310","Text":"to be in y not in x. Y equals minus b,"},{"Start":"09:34.310 ","End":"09:40.895","Text":"that\u0027s plus 3 plus or minus the square root of b^2 is"},{"Start":"09:40.895 ","End":"09:45.300","Text":"3^2 minus 4ac is minus"},{"Start":"09:45.300 ","End":"09:51.930","Text":"4 times 1 times c is minus 10."},{"Start":"09:51.930 ","End":"09:56.890","Text":"Then over to a is 2 times 1."},{"Start":"09:56.890 ","End":"10:01.015","Text":"What we get under the square root sign, let\u0027s see,"},{"Start":"10:01.015 ","End":"10:04.730","Text":"4 times 1 times 10 is 40, but it\u0027s minus,"},{"Start":"10:04.730 ","End":"10:09.320","Text":"minus, so is plus 40 plus 9 is 49."},{"Start":"10:09.320 ","End":"10:11.990","Text":"I\u0027ll make a note at the side that the square root of 49,"},{"Start":"10:11.990 ","End":"10:13.625","Text":"we\u0027ve seen this many times,"},{"Start":"10:13.625 ","End":"10:16.775","Text":"is 7, because 7 times 7 is 49."},{"Start":"10:16.775 ","End":"10:23.180","Text":"What we get is 3 plus or minus 7, 2 times 1."},{"Start":"10:23.180 ","End":"10:26.525","Text":"This is a time, this is not 21 over 2."},{"Start":"10:26.525 ","End":"10:28.429","Text":"If I take the plus,"},{"Start":"10:28.429 ","End":"10:32.034","Text":"then I get 3 plus 7 is 10,"},{"Start":"10:32.034 ","End":"10:34.275","Text":"10 over 2 is 5."},{"Start":"10:34.275 ","End":"10:41.965","Text":"The other way, 3 minus 7 is minus 4 over 2 is minus 2."},{"Start":"10:41.965 ","End":"10:47.495","Text":"But remember that we said that y has to be"},{"Start":"10:47.495 ","End":"10:54.050","Text":"positive because we need it to be the square root of a number."},{"Start":"10:54.050 ","End":"10:55.640","Text":"Square root is always positive,"},{"Start":"10:55.640 ","End":"10:59.000","Text":"so I have to rule this one out."},{"Start":"10:59.000 ","End":"11:01.540","Text":"We get that y is 5,"},{"Start":"11:01.540 ","End":"11:07.580","Text":"and if y is 5 and x is y^2,"},{"Start":"11:07.580 ","End":"11:11.135","Text":"then we get x is 25,"},{"Start":"11:11.135 ","End":"11:15.215","Text":"and this is the solution."},{"Start":"11:15.215 ","End":"11:19.230","Text":"The only one. We\u0027re done here."}],"ID":8125},{"Watched":false,"Name":"Exercise 11","Duration":"8m 9s","ChapterTopicVideoID":8033,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8033.jpeg","UploadDate":"2020-09-30T14:49:34.4130000","DurationForVideoObject":"PT8M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.475","Text":"Let\u0027s start. We have a couple of"},{"Start":"00:02.475 ","End":"00:08.445","Text":"exponential equations to solve and let\u0027s start with the first one."},{"Start":"00:08.445 ","End":"00:14.295","Text":"I\u0027ve already brought the table of formulas for exponents,"},{"Start":"00:14.295 ","End":"00:18.375","Text":"so this is a bit different than previous ones."},{"Start":"00:18.375 ","End":"00:21.320","Text":"One way of doing it is to bring"},{"Start":"00:21.320 ","End":"00:24.620","Text":"this over to the other side and then do a division, I\u0027ll show you."},{"Start":"00:24.620 ","End":"00:30.110","Text":"Write it as 192 times 3^2x equals this on"},{"Start":"00:30.110 ","End":"00:36.420","Text":"the other side becomes plus 27 times 4^(2x plus 1)."},{"Start":"00:36.420 ","End":"00:39.030","Text":"Then if we divide by,"},{"Start":"00:39.030 ","End":"00:41.070","Text":"let\u0027s say the right-hand side,"},{"Start":"00:41.070 ","End":"00:48.149","Text":"we\u0027ll just get 1 on the right and we\u0027ll get 192 times 3^2x divided"},{"Start":"00:48.149 ","End":"00:56.230","Text":"by 27 times 4^(2x plus 1) is equal to 1."},{"Start":"00:56.230 ","End":"01:00.050","Text":"I\u0027m trying to get a single exponent here."},{"Start":"01:00.050 ","End":"01:09.529","Text":"I\u0027d like to be able to use this formula with 3/4 to the power of something."},{"Start":"01:09.529 ","End":"01:12.875","Text":"Now, here I have a 2x and here I have 2x plus 1,"},{"Start":"01:12.875 ","End":"01:14.585","Text":"but that\u0027s not a problem."},{"Start":"01:14.585 ","End":"01:23.640","Text":"I can write 4^(2x plus 1) as 4^2x times 4."},{"Start":"01:23.640 ","End":"01:27.485","Text":"This is because of this formula here,"},{"Start":"01:27.485 ","End":"01:29.255","Text":"where n is 1,"},{"Start":"01:29.255 ","End":"01:32.075","Text":"and so I don\u0027t need to write 4^1."},{"Start":"01:32.075 ","End":"01:41.970","Text":"What I get here is a 192 times 3^2x over 27,"},{"Start":"01:41.970 ","End":"01:47.310","Text":"and the 4 I\u0027ll put here, times 4^2x."},{"Start":"01:47.310 ","End":"01:49.400","Text":"I\u0027m deliberately writing it like this,"},{"Start":"01:49.400 ","End":"01:52.340","Text":"numbers here and the exponents here,"},{"Start":"01:52.340 ","End":"01:55.540","Text":"and this is equal to 1."},{"Start":"01:55.540 ","End":"01:58.160","Text":"Now, I can do some number computation."},{"Start":"01:58.160 ","End":"02:00.065","Text":"Let\u0027s see if something cancels."},{"Start":"02:00.065 ","End":"02:06.035","Text":"Another 4 goes into 192 48 times."},{"Start":"02:06.035 ","End":"02:11.585","Text":"But we can still keep going because this and this are divisible by 3."},{"Start":"02:11.585 ","End":"02:16.770","Text":"If I take the 27 and divide it by 3 and the 48"},{"Start":"02:16.770 ","End":"02:21.630","Text":"divided by 3 here I will have 16 and here I\u0027ll have 9."},{"Start":"02:21.630 ","End":"02:24.169","Text":"Now, write this as a product of two fractions,"},{"Start":"02:24.169 ","End":"02:28.850","Text":"so write the 16/9 bits separately"},{"Start":"02:28.850 ","End":"02:34.460","Text":"and the 3^2x over 4^2x here."},{"Start":"02:34.460 ","End":"02:36.260","Text":"This equals 1."},{"Start":"02:36.260 ","End":"02:40.310","Text":"Now I want to bring this fraction to the other side,"},{"Start":"02:40.310 ","End":"02:45.485","Text":"so instead of multiplying on the other side it will become a divided."},{"Start":"02:45.485 ","End":"02:49.445","Text":"Dividing by a fraction is like multiplying by the reciprocal."},{"Start":"02:49.445 ","End":"02:52.175","Text":"Basically, if I continue over here,"},{"Start":"02:52.175 ","End":"03:01.110","Text":"what I get is 3^2x over 4^2x is equal to 9/16."},{"Start":"03:01.110 ","End":"03:06.810","Text":"If I take 1 and divide it by 16/9 is like multiplying by 9/16."},{"Start":"03:06.810 ","End":"03:10.245","Text":"9/16, I\u0027m looking for 3s and 4s."},{"Start":"03:10.245 ","End":"03:15.120","Text":"I see that this is 3^2 over 4^2."},{"Start":"03:15.120 ","End":"03:17.730","Text":"I\u0027m going to make double use of this."},{"Start":"03:17.730 ","End":"03:20.180","Text":"On the left-hand side,"},{"Start":"03:20.180 ","End":"03:23.000","Text":"what it means is that because this 2x and 2x are the same,"},{"Start":"03:23.000 ","End":"03:27.470","Text":"I can write it as 3/4 to the power of 2x."},{"Start":"03:27.470 ","End":"03:29.960","Text":"Here, because the 2 and the 2 are the same,"},{"Start":"03:29.960 ","End":"03:34.630","Text":"I can write this as 3/4^2."},{"Start":"03:34.630 ","End":"03:37.730","Text":"Now I\u0027m in the situation with the same base."},{"Start":"03:37.730 ","End":"03:41.270","Text":"I have base 3/4 and base 3/4."},{"Start":"03:41.270 ","End":"03:44.480","Text":"That means that I can compare the exponents,"},{"Start":"03:44.480 ","End":"03:54.454","Text":"so I say that 2x equals 2 and that just gives me that x=1 and that\u0027s the answer."},{"Start":"03:54.454 ","End":"03:57.935","Text":"Let\u0027s do a quick check and see if that fits."},{"Start":"03:57.935 ","End":"04:01.800","Text":"I have room here, so I get 192."},{"Start":"04:01.800 ","End":"04:09.240","Text":"I\u0027m looking here, times 3^2 times x is just 2 times 1 is"},{"Start":"04:09.240 ","End":"04:18.615","Text":"2 minus 27 times 4^(2x plus 1) is 3."},{"Start":"04:18.615 ","End":"04:22.909","Text":"I\u0027ll just keep working at the left-hand side and see if I get to 0."},{"Start":"04:22.909 ","End":"04:30.010","Text":"This equals, let\u0027s say 192 times 9 minus"},{"Start":"04:30.010 ","End":"04:37.895","Text":"27 times 4 times 4 times 4 is 64 and this equals."},{"Start":"04:37.895 ","End":"04:40.595","Text":"If I use a calculator,"},{"Start":"04:40.595 ","End":"04:43.690","Text":"I can say 192 times 9."},{"Start":"04:43.690 ","End":"04:52.940","Text":"I might get 1,728 and if I do 27 times 64, I checked it."},{"Start":"04:52.940 ","End":"04:56.600","Text":"I also get 1,728,"},{"Start":"04:56.600 ","End":"05:01.795","Text":"which equals 0, which is this right-hand side."},{"Start":"05:01.795 ","End":"05:06.215","Text":"So x=1 is verified as the solution."},{"Start":"05:06.215 ","End":"05:10.535","Text":"Now let\u0027s go on to part b and I\u0027ll take the formula with me."},{"Start":"05:10.535 ","End":"05:12.559","Text":"Scroll down."},{"Start":"05:12.559 ","End":"05:15.320","Text":"Part b is fairly similar to part a."},{"Start":"05:15.320 ","End":"05:17.525","Text":"We\u0027re going to use the same technique."},{"Start":"05:17.525 ","End":"05:22.730","Text":"First thing I\u0027ll do is to bring the minus over to"},{"Start":"05:22.730 ","End":"05:29.140","Text":"the other side and have it like this and then divide by the right-hand side,"},{"Start":"05:29.140 ","End":"05:31.495","Text":"so I get 1 on the right-hand side."},{"Start":"05:31.495 ","End":"05:39.430","Text":"I get here 16 times 3^(x^2 plus x) over"},{"Start":"05:39.430 ","End":"05:47.885","Text":"9 times 4^(x^2 plus x) from here is equal to 1."},{"Start":"05:47.885 ","End":"05:51.080","Text":"I can separate this fraction as"},{"Start":"05:51.080 ","End":"05:54.320","Text":"this bit times this bit because of the product of fractions,"},{"Start":"05:54.320 ","End":"05:57.895","Text":"the way it works. I\u0027ve got 16/9."},{"Start":"05:57.895 ","End":"06:00.120","Text":"In fact, I\u0027m going to do two steps in 1."},{"Start":"06:00.120 ","End":"06:04.680","Text":"I\u0027m going to bring the 16/9 to the other side is 9/16."},{"Start":"06:04.680 ","End":"06:14.475","Text":"I\u0027ve got 3^(x^2 plus x) over 4^(x^2 plus x) also."},{"Start":"06:14.475 ","End":"06:16.865","Text":"Notice the same exponent, this will be useful,"},{"Start":"06:16.865 ","End":"06:24.525","Text":"is equal to 1/16 over 9, which is 9/16."},{"Start":"06:24.525 ","End":"06:30.395","Text":"I can\u0027t help but noticing that this is 3^2 over 4^2."},{"Start":"06:30.395 ","End":"06:35.435","Text":"Now we\u0027re going to use this formula both on the left and on the right."},{"Start":"06:35.435 ","End":"06:43.130","Text":"The left-hand side becomes 3/4 to this common exponent x^2 plus x."},{"Start":"06:43.130 ","End":"06:48.575","Text":"On this side, we\u0027ll get 3/4^2 because it\u0027s the same 2,"},{"Start":"06:48.575 ","End":"06:50.500","Text":"so we can use this formula."},{"Start":"06:50.500 ","End":"06:52.895","Text":"Now we have the same base,"},{"Start":"06:52.895 ","End":"06:56.135","Text":"I mean 3/4 here and 3/4 here."},{"Start":"06:56.135 ","End":"07:03.320","Text":"We compare the exponents and say x^2 plus x equals 2 or in other words,"},{"Start":"07:03.320 ","End":"07:10.310","Text":"x^2 plus x minus 2 equals 0 and this is a quadratic equation."},{"Start":"07:10.310 ","End":"07:12.230","Text":"We know how to solve that."},{"Start":"07:12.230 ","End":"07:20.090","Text":"We have that x is equal to minus b plus or minus the square root of b^2"},{"Start":"07:20.090 ","End":"07:28.770","Text":"minus 4 times a times c. This is all over 2a,"},{"Start":"07:28.770 ","End":"07:31.215","Text":"which is twice 1 which is 2."},{"Start":"07:31.215 ","End":"07:33.270","Text":"What we get, let\u0027s see,"},{"Start":"07:33.270 ","End":"07:36.420","Text":"4 times 2 is 8 and we have minus, minus,"},{"Start":"07:36.420 ","End":"07:45.120","Text":"so we get minus 1 plus or minus the square root of 9/2."},{"Start":"07:45.120 ","End":"07:47.480","Text":"The square root of 9, of course is 3."},{"Start":"07:47.480 ","End":"07:49.610","Text":"I\u0027ll just make a note of that here."},{"Start":"07:49.610 ","End":"07:53.920","Text":"Our two solutions become minus 1 plus 3."},{"Start":"07:53.920 ","End":"08:02.265","Text":"That\u0027s 2/2 is 1 and minus 1 minus 3 is minus 4/2 is minus 2."},{"Start":"08:02.265 ","End":"08:04.350","Text":"These are my two solutions for x,"},{"Start":"08:04.350 ","End":"08:05.610","Text":"could be 1,"},{"Start":"08:05.610 ","End":"08:07.275","Text":"could be minus 2."},{"Start":"08:07.275 ","End":"08:09.370","Text":"We\u0027re done."}],"ID":8126},{"Watched":false,"Name":"Exercise 12","Duration":"2m 37s","ChapterTopicVideoID":8034,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8034.jpeg","UploadDate":"2020-09-30T13:46:35.9970000","DurationForVideoObject":"PT2M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"In this exercise, we\u0027re given a pair of exponential equations to solve."},{"Start":"00:04.710 ","End":"00:07.620","Text":"Let\u0027s start right away on the first one."},{"Start":"00:07.620 ","End":"00:12.360","Text":"Notice that in this one I have 4^x,"},{"Start":"00:12.360 ","End":"00:14.295","Text":"both here and here."},{"Start":"00:14.295 ","End":"00:18.255","Text":"It seems straightforward you have to collect like terms on the left."},{"Start":"00:18.255 ","End":"00:21.030","Text":"I have 2 apples plus 5 apples."},{"Start":"00:21.030 ","End":"00:23.760","Text":"Let\u0027s say you have 2 of these and 5 of these,"},{"Start":"00:23.760 ","End":"00:27.270","Text":"so altogether I have 2 plus 5 is 7 of these."},{"Start":"00:27.270 ","End":"00:32.160","Text":"I have 7 times 4^x is 112."},{"Start":"00:32.160 ","End":"00:35.645","Text":"Now let\u0027s just divide both sides by 7."},{"Start":"00:35.645 ","End":"00:45.799","Text":"I get 4^x is 112 divided by 7 is just 16 on the calculator or long division."},{"Start":"00:45.799 ","End":"00:49.640","Text":"Now I want to get both sides to be with the same base,"},{"Start":"00:49.640 ","End":"00:53.750","Text":"so 4^x, we know that 16 is 4^2."},{"Start":"00:53.750 ","End":"00:55.085","Text":"We\u0027ve seen it before."},{"Start":"00:55.085 ","End":"00:59.570","Text":"Now we have an expression where we have the same base 4 on both sides,"},{"Start":"00:59.570 ","End":"01:03.950","Text":"and so we compare the exponents and the answer is that x=2,"},{"Start":"01:03.950 ","End":"01:05.975","Text":"and that\u0027s all there is to it."},{"Start":"01:05.975 ","End":"01:08.855","Text":"Let\u0027s get onto the next exercise."},{"Start":"01:08.855 ","End":"01:11.720","Text":"Part b, similar thing here."},{"Start":"01:11.720 ","End":"01:15.060","Text":"We have 5x and we have 5x,"},{"Start":"01:15.060 ","End":"01:17.865","Text":"so we collect like terms."},{"Start":"01:17.865 ","End":"01:21.090","Text":"We have 3 of these things minus 1 of this,"},{"Start":"01:21.090 ","End":"01:23.535","Text":"so 3 minus 1 is 2."},{"Start":"01:23.535 ","End":"01:33.290","Text":"The left side is just 2 times 5^x and this equals 2 over 25,"},{"Start":"01:33.290 ","End":"01:35.675","Text":"dividing both sides by 2,"},{"Start":"01:35.675 ","End":"01:41.130","Text":"I\u0027ve got 5^x is 1 over 25."},{"Start":"01:41.130 ","End":"01:45.815","Text":"Let\u0027s see if we can manipulate the right-hand side to be 5 to the power of something."},{"Start":"01:45.815 ","End":"01:49.475","Text":"Well, this is equal to 1 over 5^2."},{"Start":"01:49.475 ","End":"01:51.475","Text":"That\u0027s for sure."},{"Start":"01:51.475 ","End":"01:54.860","Text":"If you have 1 over an exponent,"},{"Start":"01:54.860 ","End":"01:56.960","Text":"then it\u0027s a negative exponent."},{"Start":"01:56.960 ","End":"02:01.645","Text":"This is equal to 5 to the power of minus 2."},{"Start":"02:01.645 ","End":"02:04.460","Text":"In general, I didn\u0027t the formula sheet,"},{"Start":"02:04.460 ","End":"02:11.766","Text":"but a to the power of minus n is 1 over a to the n is one of the formulas."},{"Start":"02:11.766 ","End":"02:14.250","Text":"Here, if I have n equals 2,"},{"Start":"02:14.250 ","End":"02:15.690","Text":"and a is 5,"},{"Start":"02:15.690 ","End":"02:17.850","Text":"then I have 5 to the minus 2."},{"Start":"02:17.850 ","End":"02:23.985","Text":"Now I have the 5^x=5 to the minus 2 same base,"},{"Start":"02:23.985 ","End":"02:26.130","Text":"so compare the exponents."},{"Start":"02:26.130 ","End":"02:30.360","Text":"So x equals minus 2, and we\u0027re done."},{"Start":"02:30.360 ","End":"02:32.310","Text":"In part a we had 2,"},{"Start":"02:32.310 ","End":"02:34.455","Text":"in part b we have minus 2."},{"Start":"02:34.455 ","End":"02:37.570","Text":"That\u0027s all. We are done."}],"ID":8127},{"Watched":false,"Name":"Exercise 13","Duration":"8m 18s","ChapterTopicVideoID":8035,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8035.jpeg","UploadDate":"2020-09-30T13:55:16.9530000","DurationForVideoObject":"PT8M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.950","Text":"This exercise has 2 parts,"},{"Start":"00:01.950 ","End":"00:04.980","Text":"each of them is an exponential equation."},{"Start":"00:04.980 ","End":"00:09.855","Text":"I have a formula sheet so to speak here."},{"Start":"00:09.855 ","End":"00:12.090","Text":"Let\u0027s start with the first one."},{"Start":"00:12.090 ","End":"00:15.480","Text":"In the first one, I see very similar things,"},{"Start":"00:15.480 ","End":"00:17.485","Text":"2^x, 2^x plus 1."},{"Start":"00:17.485 ","End":"00:20.350","Text":"If I could just get it all in terms of 2^x,"},{"Start":"00:20.350 ","End":"00:24.810","Text":"and so I use this formula here, 2^x plus 1."},{"Start":"00:24.810 ","End":"00:33.015","Text":"I can use this formula and write it as 2^x times 2^1,"},{"Start":"00:33.015 ","End":"00:35.085","Text":"that\u0027s the right-hand side of this,"},{"Start":"00:35.085 ","End":"00:39.015","Text":"plus 2^x equals 48."},{"Start":"00:39.015 ","End":"00:42.735","Text":"Now, 2^1 is 2."},{"Start":"00:42.735 ","End":"00:46.740","Text":"If I look at 2^x as a common factor,"},{"Start":"00:46.740 ","End":"00:48.330","Text":"in fact, I\u0027ll just do it that way,"},{"Start":"00:48.330 ","End":"00:50.960","Text":"I\u0027ll take 2^x outside the brackets."},{"Start":"00:50.960 ","End":"00:54.140","Text":"Here I have 2^1 which is 2 of them,"},{"Start":"00:54.140 ","End":"00:59.550","Text":"and here I have 1 of them so it\u0027s plus 1 is 48."},{"Start":"00:59.550 ","End":"01:01.665","Text":"Now this thing is 3."},{"Start":"01:01.665 ","End":"01:04.545","Text":"If I divide both sides by 3,"},{"Start":"01:04.545 ","End":"01:08.910","Text":"I\u0027ll get 2^x equals 16,"},{"Start":"01:08.910 ","End":"01:12.075","Text":"and 16 we know is 2^4,"},{"Start":"01:12.075 ","End":"01:15.705","Text":"2 times 2 times 2 times 2 is 16."},{"Start":"01:15.705 ","End":"01:20.025","Text":"Here we have an expression with the same base,"},{"Start":"01:20.025 ","End":"01:28.190","Text":"so we can equate the exponents and get straightaway that x equals 4."},{"Start":"01:28.190 ","End":"01:32.280","Text":"Here\u0027s Part b. In Part b,"},{"Start":"01:32.280 ","End":"01:37.040","Text":"it\u0027s a little bit more complicated but still what we should be"},{"Start":"01:37.040 ","End":"01:42.125","Text":"able to recognize is that the base we want to choose is 6."},{"Start":"01:42.125 ","End":"01:45.470","Text":"Because we have 6 here and we have 36,"},{"Start":"01:45.470 ","End":"01:47.420","Text":"which is 6^2,"},{"Start":"01:47.420 ","End":"01:52.740","Text":"so I\u0027m going to try and write everything in terms of 6. Let\u0027s see."},{"Start":"01:54.080 ","End":"02:00.060","Text":"First of all, the external thing is the x plus 1 root,"},{"Start":"02:00.060 ","End":"02:02.535","Text":"so I want to use this formula."},{"Start":"02:02.535 ","End":"02:05.535","Text":"n is going to be like x plus 1 here."},{"Start":"02:05.535 ","End":"02:12.420","Text":"Let\u0027s first of all take care of that root and write 36^0.5x minus"},{"Start":"02:12.420 ","End":"02:22.080","Text":"1^1 over n is 1 over x plus 1."},{"Start":"02:22.080 ","End":"02:31.105","Text":"Now we have this over 36 and this is 6^2 and the square root of 6."},{"Start":"02:31.105 ","End":"02:32.600","Text":"We\u0027ve seen this often,"},{"Start":"02:32.600 ","End":"02:34.160","Text":"but I\u0027ll just highlight it."},{"Start":"02:34.160 ","End":"02:37.955","Text":"It means that we have 6^0.5,"},{"Start":"02:37.955 ","End":"02:40.630","Text":"and so let\u0027s continue."},{"Start":"02:40.630 ","End":"02:46.065","Text":"Now this 36 is 6^2."},{"Start":"02:46.065 ","End":"02:49.830","Text":"I need more space, I brought the formula down."},{"Start":"02:49.990 ","End":"02:52.640","Text":"Let\u0027s continue."},{"Start":"02:52.640 ","End":"02:54.875","Text":"I\u0027ll do a couple of things."},{"Start":"02:54.875 ","End":"02:58.920","Text":"Why don\u0027t I bring this 6^2 to the other side?"},{"Start":"02:58.920 ","End":"03:02.415","Text":"Multiply both sides and I\u0027ll get rid of that fraction."},{"Start":"03:02.415 ","End":"03:09.200","Text":"Already, I can tell you that on the right we have 6^0.5 times 6^2."},{"Start":"03:09.200 ","End":"03:13.760","Text":"It\u0027s just like bringing a denominator to the other side and making it a numerator,"},{"Start":"03:13.760 ","End":"03:16.115","Text":"division becomes a multiplication."},{"Start":"03:16.115 ","End":"03:19.905","Text":"This is the hard part, well, seemingly."},{"Start":"03:19.905 ","End":"03:22.815","Text":"36 is"},{"Start":"03:22.815 ","End":"03:30.380","Text":"(6^2)^0.5x-1"},{"Start":"03:30.380 ","End":"03:36.470","Text":"and all this ^1/x+1."},{"Start":"03:36.470 ","End":"03:38.660","Text":"Now we\u0027ve already encountered this kind of thing."},{"Start":"03:38.660 ","End":"03:40.560","Text":"We can actually do it in one step."},{"Start":"03:40.560 ","End":"03:43.370","Text":"When we have a power of a power of a power,"},{"Start":"03:43.370 ","End":"03:45.349","Text":"then we can take this formula,"},{"Start":"03:45.349 ","End":"03:49.490","Text":"which talks about the exponent of an exponent and how we multiply the exponents."},{"Start":"03:49.490 ","End":"03:51.830","Text":"We can do 3 of them not just 2."},{"Start":"03:51.830 ","End":"03:54.955","Text":"We can say that the left-hand side is"},{"Start":"03:54.955 ","End":"04:04.645","Text":"6^2 x 0.5x-1 x 1/x+1."},{"Start":"04:04.645 ","End":"04:07.190","Text":"Doing it once and then doing it again,"},{"Start":"04:07.190 ","End":"04:09.890","Text":"let\u0027s do the multiplication all in one go."},{"Start":"04:09.890 ","End":"04:14.209","Text":"On this side, we can use this formula"},{"Start":"04:14.209 ","End":"04:18.635","Text":"again for the addition we have from the right-to-left."},{"Start":"04:18.635 ","End":"04:22.355","Text":"This becomes 6^1.5 + 2."},{"Start":"04:22.355 ","End":"04:25.810","Text":"1.5 + 2 is just 2.5."},{"Start":"04:25.810 ","End":"04:30.905","Text":"Now we\u0027ve got to the situation which we like, is when we have, the same base,"},{"Start":"04:30.905 ","End":"04:34.450","Text":"an exponent on both sides with the same base,"},{"Start":"04:34.450 ","End":"04:36.615","Text":"and I mean base 6, of course."},{"Start":"04:36.615 ","End":"04:39.120","Text":"We just throw out the bases,"},{"Start":"04:39.120 ","End":"04:42.255","Text":"so just ignore them and compare the exponents."},{"Start":"04:42.255 ","End":"04:52.705","Text":"What we get is 2( 0.5x-1) x 1/x+1 = 2.5."},{"Start":"04:52.705 ","End":"04:55.475","Text":"Now it\u0027s linear equations."},{"Start":"04:55.475 ","End":"04:58.775","Text":"Not quite, it\u0027s an equation with fractions."},{"Start":"04:58.775 ","End":"05:02.085","Text":"Let\u0027s do it with a common denominator."},{"Start":"05:02.085 ","End":"05:08.080","Text":"Well, let\u0027s write the 2.5 as 5/2,"},{"Start":"05:08.080 ","End":"05:10.835","Text":"so we can see it as a fraction."},{"Start":"05:10.835 ","End":"05:16.355","Text":"Now what we can do is multiply both sides by x+1 x 2."},{"Start":"05:16.355 ","End":"05:17.735","Text":"Just write that."},{"Start":"05:17.735 ","End":"05:23.580","Text":"We\u0027re multiplying by x+1 and multiplying by 2."},{"Start":"05:23.580 ","End":"05:24.905","Text":"I\u0027ll put the 2 here."},{"Start":"05:24.905 ","End":"05:31.220","Text":"What we have here is the x+1 will cancel and we still have to multiply this by 2."},{"Start":"05:31.220 ","End":"05:33.440","Text":"Here on the right-hand side,"},{"Start":"05:33.440 ","End":"05:38.230","Text":"we\u0027ll have to just multiply by x+1."},{"Start":"05:38.230 ","End":"05:43.710","Text":"Continuing over here, we get 2 x 2"},{"Start":"05:43.710 ","End":"05:53.985","Text":"x 0.5x-1 = 5 x x+1."},{"Start":"05:53.985 ","End":"05:57.405","Text":"Let\u0027s multiply out 2 x 2 is 4,"},{"Start":"05:57.405 ","End":"06:03.975","Text":"so 4 x 0.5x is 2x because 4 x 0.5 is"},{"Start":"06:03.975 ","End":"06:07.780","Text":"half of 4 is 2-4"},{"Start":"06:08.030 ","End":"06:15.060","Text":"= 5x + 5."},{"Start":"06:15.060 ","End":"06:16.785","Text":"Now x is to the left,"},{"Start":"06:16.785 ","End":"06:18.315","Text":"numbers to the right."},{"Start":"06:18.315 ","End":"06:23.520","Text":"2x-5x=5+4."},{"Start":"06:23.520 ","End":"06:29.310","Text":"So -3x=9 and %"},{"Start":"06:29.310 ","End":"06:34.940","Text":"-3, x is -9/3-3."},{"Start":"06:34.940 ","End":"06:38.870","Text":"Now it would appear that we\u0027re done and we would just say x =-3,"},{"Start":"06:38.870 ","End":"06:40.070","Text":"but not so fast."},{"Start":"06:40.070 ","End":"06:43.820","Text":"There are certain restrictions and I didn\u0027t point them out earlier."},{"Start":"06:43.820 ","End":"06:45.260","Text":"I didn\u0027t want to confuse things,"},{"Start":"06:45.260 ","End":"06:47.015","Text":"but we have to relate to them."},{"Start":"06:47.015 ","End":"06:49.925","Text":"For one thing, when we have a fraction,"},{"Start":"06:49.925 ","End":"06:53.405","Text":"I should have mentioned that the denominator can\u0027t be 0,"},{"Start":"06:53.405 ","End":"07:00.720","Text":"so here I could have mentioned that x+1 must not be 0 and indeed x+1 is not 0."},{"Start":"07:00.720 ","End":"07:02.360","Text":"But it\u0027s more than that."},{"Start":"07:02.360 ","End":"07:06.485","Text":"This comes from the fact that this is the x+1 root."},{"Start":"07:06.485 ","End":"07:09.830","Text":"I have an expression with x+1 here."},{"Start":"07:09.830 ","End":"07:12.785","Text":"Now, when you take a radical or a root,"},{"Start":"07:12.785 ","End":"07:18.274","Text":"for example here, that root must be a positive whole number."},{"Start":"07:18.274 ","End":"07:20.435","Text":"I don\u0027t stress it each time,"},{"Start":"07:20.435 ","End":"07:23.810","Text":"but here it\u0027s important and it has to be a positive whole number like 1,"},{"Start":"07:23.810 ","End":"07:27.680","Text":"2, 3, 4, and so on and certainly not 0."},{"Start":"07:27.680 ","End":"07:30.150","Text":"It\u0027s more than just that it\u0027s not 0,"},{"Start":"07:30.150 ","End":"07:33.913","Text":"we have to have that x+1 has to be either 1,"},{"Start":"07:33.913 ","End":"07:37.410","Text":"2, 3, 4, etc."},{"Start":"07:37.410 ","End":"07:42.015","Text":"Here, x+1 is -2,"},{"Start":"07:42.015 ","End":"07:45.997","Text":"which is not good because x+1 has to be 1,"},{"Start":"07:45.997 ","End":"07:50.240","Text":"2, 3, 4, 5, so actually I have to rule this solution out."},{"Start":"07:50.240 ","End":"07:52.030","Text":"This is not good."},{"Start":"07:52.030 ","End":"07:55.145","Text":"Actually there is no solution."},{"Start":"07:55.145 ","End":"07:58.475","Text":"There was only one possible solution that could have been,"},{"Start":"07:58.475 ","End":"08:00.935","Text":"but it\u0027s illegal because we can\u0027t take,"},{"Start":"08:00.935 ","End":"08:06.065","Text":"by definition, the minus 2th root or something."},{"Start":"08:06.065 ","End":"08:08.127","Text":"A root always is 1,"},{"Start":"08:08.127 ","End":"08:09.215","Text":"2, 3, 4, 5,"},{"Start":"08:09.215 ","End":"08:11.885","Text":"certainly not 0 or negative,"},{"Start":"08:11.885 ","End":"08:17.640","Text":"so no solution that can happen. We are done."}],"ID":8128},{"Watched":false,"Name":"Exercise 14","Duration":"5m 1s","ChapterTopicVideoID":8036,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8036.jpeg","UploadDate":"2020-09-30T13:58:14.9170000","DurationForVideoObject":"PT5M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"We have 2 equations to solve,"},{"Start":"00:02.640 ","End":"00:04.920","Text":"each of them involving exponents."},{"Start":"00:04.920 ","End":"00:08.445","Text":"I have the formula page sheet here,"},{"Start":"00:08.445 ","End":"00:10.440","Text":"and let\u0027s start with the first one."},{"Start":"00:10.440 ","End":"00:14.835","Text":"We see there\u0027s 2^x here and 2^x plus 3 here."},{"Start":"00:14.835 ","End":"00:21.269","Text":"We can get this all in terms of 2^x if we use some of the rules here, for example,"},{"Start":"00:21.269 ","End":"00:24.840","Text":"this one, I can use this formula for"},{"Start":"00:24.840 ","End":"00:29.055","Text":"the sum of exponents and convert the sum to a product."},{"Start":"00:29.055 ","End":"00:39.270","Text":"This becomes 2^x times 2^3 minus 2^x =28."},{"Start":"00:39.270 ","End":"00:43.860","Text":"Now I can collect like terms for 2^x."},{"Start":"00:43.860 ","End":"00:47.235","Text":"This is like 2^3 is 8,"},{"Start":"00:47.235 ","End":"00:49.560","Text":"and this is minus 1,"},{"Start":"00:49.560 ","End":"00:52.150","Text":"we actually have 8 minus 1, 2^x."},{"Start":"00:52.150 ","End":"00:54.080","Text":"If I take outside the brackets,"},{"Start":"00:54.080 ","End":"01:00.920","Text":"I could\u0027ve done it in 1 step and said straight away that this is 7 times 2^x."},{"Start":"01:00.920 ","End":"01:04.020","Text":"Anyway, probably better to do it the long way."},{"Start":"01:04.060 ","End":"01:09.785","Text":"The right-hand side stays 28."},{"Start":"01:09.785 ","End":"01:19.910","Text":"Now is the point at which we can divide both sides by 7 and get 2^x is 28/7, which is 4."},{"Start":"01:19.910 ","End":"01:23.640","Text":"4, of course, is 2^2."},{"Start":"01:23.640 ","End":"01:25.180","Text":"Now we have the same base,"},{"Start":"01:25.180 ","End":"01:29.500","Text":"so we can compare exponents and get that x=2."},{"Start":"01:29.500 ","End":"01:33.430","Text":"That\u0027s part a. From time to time,"},{"Start":"01:33.430 ","End":"01:36.400","Text":"I like to check my solution. Let\u0027s do it."},{"Start":"01:36.400 ","End":"01:39.550","Text":"What I\u0027ll do is substitute x=2 in"},{"Start":"01:39.550 ","End":"01:42.865","Text":"the left-hand side and see if we can get to the right-hand side."},{"Start":"01:42.865 ","End":"01:46.205","Text":"2^, and now x is 2 here,"},{"Start":"01:46.205 ","End":"01:50.960","Text":"2 plus 3 minus 2^2,"},{"Start":"01:50.960 ","End":"01:53.310","Text":"and just keep going."},{"Start":"01:53.310 ","End":"01:59.895","Text":"This is equal to 2^2 plus 3 is 5 minus 2^2."},{"Start":"01:59.895 ","End":"02:02.775","Text":"2^5 is 32."},{"Start":"02:02.775 ","End":"02:05.880","Text":"We know this. 2^2 is 4."},{"Start":"02:05.880 ","End":"02:07.980","Text":"This equals 28,"},{"Start":"02:07.980 ","End":"02:10.545","Text":"which is indeed the right-hand side."},{"Start":"02:10.545 ","End":"02:13.969","Text":"X=2 is a verified solution."},{"Start":"02:13.969 ","End":"02:17.410","Text":"Now let\u0027s go and do part b."},{"Start":"02:17.410 ","End":"02:20.030","Text":"We have something similar here."},{"Start":"02:20.030 ","End":"02:24.260","Text":"We also have everything\u0027s based on 4^x with a variation."},{"Start":"02:24.260 ","End":"02:28.020","Text":"But let\u0027s try and express everything in terms of 4^x."},{"Start":"02:28.020 ","End":"02:31.745","Text":"I\u0027m going to use this formula again twice here and here,"},{"Start":"02:31.745 ","End":"02:34.415","Text":"and just get a bit more space there."},{"Start":"02:34.415 ","End":"02:37.175","Text":"We have 1.5."},{"Start":"02:37.175 ","End":"02:44.760","Text":"Now 4^x plus 2 as I said using this formula is 4^x times 4^2."},{"Start":"02:44.760 ","End":"02:52.695","Text":"Then 4^x plus 1 is 4^x, 4^1."},{"Start":"02:52.695 ","End":"02:55.830","Text":"I\u0027ll leave the right-hand side alone for a moment."},{"Start":"02:55.830 ","End":"03:00.530","Text":"Now, I think we can just do one small thing instead of 16 is 4^2."},{"Start":"03:00.530 ","End":"03:04.040","Text":"That will just show us that we\u0027re heading for a powers of 4."},{"Start":"03:04.040 ","End":"03:10.060","Text":"Okay. Let\u0027s see now, 4^2 is 16."},{"Start":"03:10.060 ","End":"03:13.520","Text":"This is a product, so I can do 16/2 is 8."},{"Start":"03:13.520 ","End":"03:15.190","Text":"This is 8."},{"Start":"03:15.190 ","End":"03:17.940","Text":"I can put it in front of the 4^x."},{"Start":"03:17.940 ","End":"03:20.355","Text":"Here I have 4^1 is 4."},{"Start":"03:20.355 ","End":"03:23.220","Text":"This is 4 times 4^x."},{"Start":"03:23.220 ","End":"03:28.920","Text":"Here we have 3/4^2."},{"Start":"03:28.920 ","End":"03:32.700","Text":"Now, 8 and 4 is 12,"},{"Start":"03:32.700 ","End":"03:41.590","Text":"so I can write this as 12 times 4^x equals 3/4^2."},{"Start":"03:42.620 ","End":"03:45.785","Text":"Now continuing over here,"},{"Start":"03:45.785 ","End":"03:48.875","Text":"we have that 4^x."},{"Start":"03:48.875 ","End":"03:50.810","Text":"If I bring the 12 to the other side,"},{"Start":"03:50.810 ","End":"04:00.630","Text":"it goes into the denominator, =3/12 times 4^2."},{"Start":"04:00.630 ","End":"04:02.790","Text":"Now let\u0027s do some canceling."},{"Start":"04:02.790 ","End":"04:07.335","Text":"3 goes into 12, 4 times."},{"Start":"04:07.335 ","End":"04:16.465","Text":"So what I get now is 4^x=1/4 times 4^2."},{"Start":"04:16.465 ","End":"04:21.855","Text":"Now I\u0027m going to use this formula again because 4 is 4^1."},{"Start":"04:21.855 ","End":"04:29.155","Text":"So I can write it as 1/4^1 plus 2 is 3."},{"Start":"04:29.155 ","End":"04:31.595","Text":"I\u0027m using this formula from right to left."},{"Start":"04:31.595 ","End":"04:35.450","Text":"Now if I use this formula and I can deal with"},{"Start":"04:35.450 ","End":"04:41.940","Text":"the denominator if I go from right to left and say that this equals 4 to the minus 3."},{"Start":"04:41.940 ","End":"04:44.795","Text":"This is 4 and then there is 3 here."},{"Start":"04:44.795 ","End":"04:47.915","Text":"Now if I look at this and this same base,"},{"Start":"04:47.915 ","End":"04:50.730","Text":"base 4 and base 4."},{"Start":"04:50.730 ","End":"04:53.000","Text":"Now I can compare the exponents."},{"Start":"04:53.000 ","End":"04:58.130","Text":"This gives me straight away that x equals minus 3."},{"Start":"04:58.130 ","End":"05:01.290","Text":"That\u0027s the answer and we\u0027re done."}],"ID":8129},{"Watched":false,"Name":"Exercise 15","Duration":"6m 7s","ChapterTopicVideoID":8037,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8037.jpeg","UploadDate":"2020-09-30T14:01:15.9330000","DurationForVideoObject":"PT6M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.640","Text":"In this exercise, we have 2 exponential equations to solve,"},{"Start":"00:05.640 ","End":"00:08.640","Text":"and I have some formulas here."},{"Start":"00:08.640 ","End":"00:11.340","Text":"I\u0027m looking at both of them and I see in both of them"},{"Start":"00:11.340 ","End":"00:14.040","Text":"I have an exponent with a subtraction here."},{"Start":"00:14.040 ","End":"00:15.420","Text":"I have x minus 1,"},{"Start":"00:15.420 ","End":"00:17.745","Text":"here I have x minus 2, here x minus 1."},{"Start":"00:17.745 ","End":"00:20.295","Text":"The main formula will be this one."},{"Start":"00:20.295 ","End":"00:22.769","Text":"Anyway, let\u0027s get started on the first."},{"Start":"00:22.769 ","End":"00:25.860","Text":"Here we have 4 and we\u0027re going to use"},{"Start":"00:25.860 ","End":"00:29.850","Text":"that formula I mentioned where we have an exponent which is a difference,"},{"Start":"00:29.850 ","End":"00:32.100","Text":"then it turns into a division."},{"Start":"00:32.100 ","End":"00:41.090","Text":"This will be 5^x/5^1 plus"},{"Start":"00:41.090 ","End":"00:45.935","Text":"5^x=0.36."},{"Start":"00:45.935 ","End":"00:48.665","Text":"We\u0027ll see what to do with this in a while."},{"Start":"00:48.665 ","End":"00:52.940","Text":"Then what I\u0027m going to do is collect like terms."},{"Start":"00:52.940 ","End":"00:55.910","Text":"I\u0027ll take 5^x outside the brackets."},{"Start":"00:55.910 ","End":"00:58.955","Text":"Basically, I want to count how many times do I have 5^x?"},{"Start":"00:58.955 ","End":"01:02.990","Text":"Well, from here, I have 4/5^1 which is 4/5."},{"Start":"01:02.990 ","End":"01:07.190","Text":"Here I have another 1 times 5^x."},{"Start":"01:07.190 ","End":"01:11.790","Text":"Altogether I have 1 and 4/5, and you know what?"},{"Start":"01:11.790 ","End":"01:16.350","Text":"Let\u0027s work in decimals since we have a decimal on the right."},{"Start":"01:16.350 ","End":"01:22.395","Text":"This will be 1 and 4/5 is 1.8 times 5^x."},{"Start":"01:22.395 ","End":"01:27.255","Text":"Here 0.36, 0.36."},{"Start":"01:27.255 ","End":"01:32.765","Text":"So 5^x, we can get by dividing this by this."},{"Start":"01:32.765 ","End":"01:37.955","Text":"It\u0027s 0.36/1.8."},{"Start":"01:37.955 ","End":"01:39.980","Text":"We can do that on the calculator,"},{"Start":"01:39.980 ","End":"01:42.395","Text":"or we can do it mentally."},{"Start":"01:42.395 ","End":"01:44.900","Text":"In any event, one way or another,"},{"Start":"01:44.900 ","End":"01:48.080","Text":"it comes out to be 0.2,"},{"Start":"01:48.080 ","End":"01:50.990","Text":"and 0.2 is very familiar,"},{"Start":"01:50.990 ","End":"01:54.550","Text":"it\u0027s 2/10, it\u0027s a 1/5."},{"Start":"01:54.550 ","End":"01:57.754","Text":"All I have to do is recall this formula."},{"Start":"01:57.754 ","End":"02:01.680","Text":"If I use it with a=5 and n=1,"},{"Start":"02:01.680 ","End":"02:05.320","Text":"then this becomes 5^ minus 1."},{"Start":"02:05.320 ","End":"02:08.330","Text":"Now, look at the beginning of the chain and the end of the chain."},{"Start":"02:08.330 ","End":"02:09.830","Text":"Here and here,"},{"Start":"02:09.830 ","End":"02:14.770","Text":"we have 2 exponents with the same base, 5."},{"Start":"02:14.770 ","End":"02:17.670","Text":"Here\u0027s a 5, and here\u0027s a 5."},{"Start":"02:17.670 ","End":"02:23.680","Text":"We compare the exponents and then we immediately get that x= minus 1,"},{"Start":"02:23.680 ","End":"02:26.315","Text":"and that\u0027s the answer to part a."},{"Start":"02:26.315 ","End":"02:27.920","Text":"Like I said, from time to time,"},{"Start":"02:27.920 ","End":"02:29.990","Text":"we\u0027ll check by substituting."},{"Start":"02:29.990 ","End":"02:31.730","Text":"Let\u0027s try that here."},{"Start":"02:31.730 ","End":"02:34.080","Text":"Put x equals minus 1 in the original,"},{"Start":"02:34.080 ","End":"02:43.550","Text":"and so what we get if we substitute x= minus 1 is 4 times 5^ minus 1,"},{"Start":"02:43.550 ","End":"02:47.760","Text":"minus 1 is minus 2 plus 5,"},{"Start":"02:47.760 ","End":"02:50.915","Text":"x is minus 1 equals."},{"Start":"02:50.915 ","End":"02:54.740","Text":"Now, I want to reach the right-hand side to see"},{"Start":"02:54.740 ","End":"02:58.715","Text":"if I do get 0.36 because I don\u0027t know this is true at the moment."},{"Start":"02:58.715 ","End":"03:00.440","Text":"Let\u0027s see what this equals."},{"Start":"03:00.440 ","End":"03:05.455","Text":"5 to the minus 2 is 1/25."},{"Start":"03:05.455 ","End":"03:09.915","Text":"If I look at this, 5 to the minus 2 is 1/5 squared is 1/25."},{"Start":"03:09.915 ","End":"03:14.760","Text":"This is 4/25 minus,"},{"Start":"03:14.760 ","End":"03:17.775","Text":"and 5 to the minus 1 is 1/5."},{"Start":"03:17.775 ","End":"03:21.850","Text":"Let\u0027s convert this to decimal."},{"Start":"03:21.860 ","End":"03:25.580","Text":"4/25, if we multiply top and bottom by 4,"},{"Start":"03:25.580 ","End":"03:27.415","Text":"we get 16/100,"},{"Start":"03:27.415 ","End":"03:31.440","Text":"so this is 0.16."},{"Start":"03:31.490 ","End":"03:35.430","Text":"Oops, I miscopied the plus with a minus."},{"Start":"03:35.430 ","End":"03:37.125","Text":"This is a plus of course,"},{"Start":"03:37.125 ","End":"03:42.090","Text":"and 1/5 we know is 0.2."},{"Start":"03:42.090 ","End":"03:44.360","Text":"If we add these 2 decimals,"},{"Start":"03:44.360 ","End":"03:47.330","Text":"what we get is 0.36,"},{"Start":"03:47.330 ","End":"03:50.815","Text":"which is what we have on the right-hand side."},{"Start":"03:50.815 ","End":"03:54.770","Text":"Now I can say that x equals minus 1 is a verified solution."},{"Start":"03:54.770 ","End":"03:57.260","Text":"I\u0027ve substituted it and it works."},{"Start":"03:57.260 ","End":"04:01.740","Text":"Let\u0027s move on to part b."},{"Start":"04:02.570 ","End":"04:05.270","Text":"We have something similar here,"},{"Start":"04:05.270 ","End":"04:07.190","Text":"except that there are 3 terms."},{"Start":"04:07.190 ","End":"04:09.710","Text":"I\u0027m going to use this formula again."},{"Start":"04:09.710 ","End":"04:19.445","Text":"We have 3 times 2^x minus 5 times 2^x over 2^2 from there."},{"Start":"04:19.445 ","End":"04:23.605","Text":"Here, 2^x over 2^1,"},{"Start":"04:23.605 ","End":"04:25.320","Text":"which is just 2,"},{"Start":"04:25.320 ","End":"04:27.390","Text":"is equal to 40."},{"Start":"04:27.390 ","End":"04:30.770","Text":"I\u0027d like to get rid of fractions."},{"Start":"04:30.770 ","End":"04:36.305","Text":"Note that this here is 4 and this here is 2."},{"Start":"04:36.305 ","End":"04:40.215","Text":"A common denominator will be 4,"},{"Start":"04:40.215 ","End":"04:44.855","Text":"so if I multiply everything by the common denominator using our usual method,"},{"Start":"04:44.855 ","End":"04:47.590","Text":"here I have to multiply by 4."},{"Start":"04:47.590 ","End":"04:50.250","Text":"Here the 4/4 cancels,"},{"Start":"04:50.250 ","End":"04:52.650","Text":"so I only have to multiply by 1."},{"Start":"04:52.650 ","End":"04:55.105","Text":"Here 4/2 is 2,"},{"Start":"04:55.105 ","End":"04:58.055","Text":"and here I have to multiply everything by 4."},{"Start":"04:58.055 ","End":"05:06.740","Text":"What I get is 4 times 3 is 12 times 2^x minus 1 times 5 is"},{"Start":"05:06.740 ","End":"05:17.420","Text":"5 times 2^x minus 2 times 2^x equals 4 times 40 is 160."},{"Start":"05:17.420 ","End":"05:21.090","Text":"Now here everything is in terms of 2^x."},{"Start":"05:21.090 ","End":"05:23.435","Text":"I think we can do this mentally."},{"Start":"05:23.435 ","End":"05:30.960","Text":"12 minus 5 minus 2 is 12 minus 7 is 5."},{"Start":"05:30.960 ","End":"05:37.025","Text":"We have 5 times 2^x equals 160."},{"Start":"05:37.025 ","End":"05:41.455","Text":"Continue over here, divide both sides by 5,"},{"Start":"05:41.455 ","End":"05:46.980","Text":"2^x is 16/5 is 32,"},{"Start":"05:46.980 ","End":"05:52.590","Text":"and 32, we\u0027ve seen this often enough, is 2^5."},{"Start":"05:52.590 ","End":"05:56.655","Text":"Now we have 2 exponents with the same base 2,"},{"Start":"05:56.655 ","End":"06:03.005","Text":"so we compare the exponents and get right away that x equals 5."},{"Start":"06:03.005 ","End":"06:05.060","Text":"This finishes part b,"},{"Start":"06:05.060 ","End":"06:07.560","Text":"and so we are done."}],"ID":8130},{"Watched":false,"Name":"Exercise 16","Duration":"7m 36s","ChapterTopicVideoID":8038,"CourseChapterTopicPlaylistID":56156,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8038.jpeg","UploadDate":"2020-09-30T14:04:45.0200000","DurationForVideoObject":"PT7M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"Here we are continuing with exponential equations."},{"Start":"00:03.420 ","End":"00:04.710","Text":"We\u0027ve got another pair."},{"Start":"00:04.710 ","End":"00:06.990","Text":"Let\u0027s start with the first one."},{"Start":"00:06.990 ","End":"00:09.780","Text":"I\u0027m looking at this and I see something a bit different,"},{"Start":"00:09.780 ","End":"00:11.820","Text":"is that I have three different bases."},{"Start":"00:11.820 ","End":"00:13.080","Text":"I have base 6,"},{"Start":"00:13.080 ","End":"00:15.210","Text":"base 3 and base 2."},{"Start":"00:15.210 ","End":"00:19.035","Text":"The question is how to get it all in terms of one base."},{"Start":"00:19.035 ","End":"00:22.140","Text":"Now, I also look at the exponents and see I have an x"},{"Start":"00:22.140 ","End":"00:25.530","Text":"minus 1 here and here and here I have an x."},{"Start":"00:25.530 ","End":"00:27.825","Text":"But supposing this was also x,"},{"Start":"00:27.825 ","End":"00:32.755","Text":"then looking at this formula, I mean,"},{"Start":"00:32.755 ","End":"00:34.550","Text":"and looking at it from right to left,"},{"Start":"00:34.550 ","End":"00:36.830","Text":"if a is 3 and b is 2,"},{"Start":"00:36.830 ","End":"00:39.160","Text":"then I could combine 3_x,"},{"Start":"00:39.160 ","End":"00:43.275","Text":"2_x and get 6_x because that b, would be 6."},{"Start":"00:43.275 ","End":"00:45.015","Text":"But I don\u0027t quite have that."},{"Start":"00:45.015 ","End":"00:49.025","Text":"Let\u0027s just manipulate it a bit until we do and then we can use this formula."},{"Start":"00:49.025 ","End":"00:51.305","Text":"Meanwhile, to get rid of the minus 1,"},{"Start":"00:51.305 ","End":"00:53.005","Text":"we\u0027ll use this formula."},{"Start":"00:53.005 ","End":"01:01.030","Text":"We start off by saying that this is 5 times 6_x over 6_1."},{"Start":"01:01.030 ","End":"01:03.270","Text":"Don\u0027t write 6_1, just write 6."},{"Start":"01:03.270 ","End":"01:09.710","Text":"Here I have 3_x times 2_x over 2_1,"},{"Start":"01:09.710 ","End":"01:12.380","Text":"from this formula, but I don\u0027t write the 1,"},{"Start":"01:12.380 ","End":"01:15.235","Text":"is equal to 72."},{"Start":"01:15.235 ","End":"01:17.930","Text":"Now I can get rid of fractions first,"},{"Start":"01:17.930 ","End":"01:19.010","Text":"or I could do this first."},{"Start":"01:19.010 ","End":"01:21.955","Text":"I want to get rid of this 3 times 2 first."},{"Start":"01:21.955 ","End":"01:25.010","Text":"Let\u0027s leave the first one as is,"},{"Start":"01:25.010 ","End":"01:31.045","Text":"but I\u0027ll just write it slightly differently as 5/6 times 6_x."},{"Start":"01:31.045 ","End":"01:36.665","Text":"Here we\u0027ll write this as minus 1/2 times this thing here."},{"Start":"01:36.665 ","End":"01:42.335","Text":"Now, this bit here is what I want to use in this formula here."},{"Start":"01:42.335 ","End":"01:47.405","Text":"I can write this as 3 times 2^x."},{"Start":"01:47.405 ","End":"01:49.310","Text":"That\u0027s the left-hand side of this,"},{"Start":"01:49.310 ","End":"01:52.955","Text":"so a is 3 and b is 2, and equals 72."},{"Start":"01:52.955 ","End":"01:55.175","Text":"I\u0027ll tell you what, let\u0027s save a step."},{"Start":"01:55.175 ","End":"01:58.610","Text":"Because everyone knows that 3 times 2 is 6."},{"Start":"01:58.610 ","End":"02:02.360","Text":"I have 6_x here and 6_x here."},{"Start":"02:02.360 ","End":"02:06.560","Text":"I could combine like terms or I could get rid of fractions."},{"Start":"02:06.560 ","End":"02:08.450","Text":"Tell you what, let\u0027s get rid of fractions."},{"Start":"02:08.450 ","End":"02:11.960","Text":"Let\u0027s see what\u0027s the common denominator for 6 and 2, obviously 6."},{"Start":"02:11.960 ","End":"02:16.350","Text":"Let\u0027s multiply both sides by 6."},{"Start":"02:16.350 ","End":"02:21.450","Text":"Then we get, multiplying by 6 here just cancels with the denominator,"},{"Start":"02:21.450 ","End":"02:24.130","Text":"we have 5 times 6_x."},{"Start":"02:24.130 ","End":"02:25.620","Text":"Multiplying by 6 here,"},{"Start":"02:25.620 ","End":"02:27.885","Text":"6 over 2 is 3."},{"Start":"02:27.885 ","End":"02:34.320","Text":"I have 3 times 6_x is equal to 72."},{"Start":"02:34.320 ","End":"02:37.755","Text":"Now 5 of these minus 3 of these."},{"Start":"02:37.755 ","End":"02:40.050","Text":"I only have 2 of these,"},{"Start":"02:40.050 ","End":"02:44.520","Text":"it\u0027s 2 times 6_x is 72."},{"Start":"02:44.520 ","End":"02:49.245","Text":"That gives us that 6_x is equal to,"},{"Start":"02:49.245 ","End":"02:52.095","Text":"divided by 2 is 36."},{"Start":"02:52.095 ","End":"02:56.655","Text":"Now 36 everyone knows is 6 times 6, which is 6^2."},{"Start":"02:56.655 ","End":"03:01.415","Text":"We\u0027ve reached the point where we have an equality with the same base."},{"Start":"03:01.415 ","End":"03:03.710","Text":"We then compare the exponents."},{"Start":"03:03.710 ","End":"03:07.250","Text":"That leaves us with x equals 2."},{"Start":"03:07.250 ","End":"03:09.215","Text":"That\u0027s the answer to part a."},{"Start":"03:09.215 ","End":"03:13.305","Text":"Let\u0027s move on to part b. Scroll down a bit."},{"Start":"03:13.305 ","End":"03:15.570","Text":"There we are."},{"Start":"03:15.570 ","End":"03:18.615","Text":"Still got the formulas here."},{"Start":"03:18.615 ","End":"03:22.650","Text":"Now we\u0027re going to tackle this one. Let\u0027s see."},{"Start":"03:22.650 ","End":"03:28.520","Text":"We have 2 and 5 in each case, but different exponents."},{"Start":"03:28.520 ","End":"03:30.965","Text":"I\u0027d like to try again and use this."},{"Start":"03:30.965 ","End":"03:35.750","Text":"Let\u0027s get rid of those extra plus 2 here and plus 1."},{"Start":"03:35.750 ","End":"03:37.870","Text":"Well, first of all, apply,"},{"Start":"03:37.870 ","End":"03:39.290","Text":"well not this formula,"},{"Start":"03:39.290 ","End":"03:42.545","Text":"it has a minus, but the similar one."},{"Start":"03:42.545 ","End":"03:44.720","Text":"When we have a plus, it\u0027s similar, with the minus,"},{"Start":"03:44.720 ","End":"03:47.480","Text":"we get division with a plus, we get multiplication."},{"Start":"03:47.480 ","End":"03:49.310","Text":"Applying it to this,"},{"Start":"03:49.310 ","End":"03:52.445","Text":"this and this, we get 2_x."},{"Start":"03:52.445 ","End":"03:56.135","Text":"Now, here we have 5_x,"},{"Start":"03:56.135 ","End":"04:00.990","Text":"5_2, 5^2 multiplied,"},{"Start":"04:00.990 ","End":"04:04.125","Text":"minus, and here we have 2_x,"},{"Start":"04:04.125 ","End":"04:10.520","Text":"2_2, and here 5_x, 5_1."},{"Start":"04:10.520 ","End":"04:12.680","Text":"Right-hand side is unchanged."},{"Start":"04:12.680 ","End":"04:17.570","Text":"Now look, we have here 2_x, 5_x,"},{"Start":"04:17.570 ","End":"04:20.090","Text":"and here, even though it\u0027s separated,"},{"Start":"04:20.090 ","End":"04:22.625","Text":"we have 2_x, 5_x."},{"Start":"04:22.625 ","End":"04:26.585","Text":"Let\u0027s count how many 2_x, 5_x we have."},{"Start":"04:26.585 ","End":"04:29.320","Text":"Or if you\u0027d like, we can take this outside the brackets."},{"Start":"04:29.320 ","End":"04:32.900","Text":"I\u0027m going to take 2_x, 5_x outside the brackets."},{"Start":"04:32.900 ","End":"04:36.110","Text":"I\u0027ll put it on the right this time, that live on the left."},{"Start":"04:36.110 ","End":"04:39.400","Text":"What we\u0027re left with is, here we have,"},{"Start":"04:39.400 ","End":"04:42.055","Text":"the leftover is 5^2,"},{"Start":"04:42.055 ","End":"04:44.070","Text":"and here we have 2 leftovers."},{"Start":"04:44.070 ","End":"04:49.960","Text":"We have 2^2 and 5^1."},{"Start":"04:49.960 ","End":"04:52.775","Text":"This is just counting how many times we have 2_x,"},{"Start":"04:52.775 ","End":"04:55.535","Text":"5_x is still equal to 50."},{"Start":"04:55.535 ","End":"04:58.280","Text":"Here is just some arithmetic."},{"Start":"04:58.280 ","End":"05:01.010","Text":"Let\u0027s see, 5^2 is 25,"},{"Start":"05:01.010 ","End":"05:04.155","Text":"2^2 times 5 is 4 times 5 is 20."},{"Start":"05:04.155 ","End":"05:05.895","Text":"This gives us 5."},{"Start":"05:05.895 ","End":"05:08.445","Text":"We have 5 times 2_x,"},{"Start":"05:08.445 ","End":"05:12.855","Text":"5_x is equal to 50."},{"Start":"05:12.855 ","End":"05:15.015","Text":"Dividing by 5,"},{"Start":"05:15.015 ","End":"05:19.855","Text":"I\u0027ve got 2_x, 5_x equals 10."},{"Start":"05:19.855 ","End":"05:21.590","Text":"Now I held off on doing something."},{"Start":"05:21.590 ","End":"05:23.345","Text":"I don\u0027t know why held off so long,"},{"Start":"05:23.345 ","End":"05:29.450","Text":"but we\u0027re going to use this formula with a is 2 and b is 5."},{"Start":"05:29.450 ","End":"05:32.255","Text":"2_x, 5_x, because it\u0027s the same x,"},{"Start":"05:32.255 ","End":"05:38.325","Text":"we can multiply 2 times 5 and say this is 10_x is equal to 10."},{"Start":"05:38.325 ","End":"05:40.065","Text":"I\u0027ll just write down at the side,"},{"Start":"05:40.065 ","End":"05:42.430","Text":"2 times 5 equals 10,"},{"Start":"05:42.430 ","End":"05:43.835","Text":"in case you didn\u0027t know that."},{"Start":"05:43.835 ","End":"05:46.610","Text":"Now we have 10^x equals 10."},{"Start":"05:46.610 ","End":"05:49.205","Text":"How do we continue from here?"},{"Start":"05:49.205 ","End":"05:52.055","Text":"Well, 10_x is equal to,"},{"Start":"05:52.055 ","End":"05:54.500","Text":"I can rewrite 10 is 10_1."},{"Start":"05:54.500 ","End":"05:59.060","Text":"Now I have my favorite situation where we have an exponent,"},{"Start":"05:59.060 ","End":"06:00.920","Text":"but the bases are equal,"},{"Start":"06:00.920 ","End":"06:04.475","Text":"and then we compare the powers, the exponents."},{"Start":"06:04.475 ","End":"06:07.160","Text":"Immediately, we get that x=1."},{"Start":"06:07.160 ","End":"06:14.570","Text":"Why don\u0027t I verify by substituting x equals 1 in the original,"},{"Start":"06:14.570 ","End":"06:16.010","Text":"we should do this from time to time."},{"Start":"06:16.010 ","End":"06:20.360","Text":"I\u0027m plugging this in here and I\u0027m continuing over here."},{"Start":"06:20.360 ","End":"06:26.945","Text":"I have 2_1 times 5_^1 plus 2"},{"Start":"06:26.945 ","End":"06:35.155","Text":"minus 2_1 plus 2 times 5^1 plus 1."},{"Start":"06:35.155 ","End":"06:39.905","Text":"What I\u0027ll do is I\u0027ll work just on the left-hand side and see if I can reach 50."},{"Start":"06:39.905 ","End":"06:41.840","Text":"This is equal to, let\u0027s see,"},{"Start":"06:41.840 ","End":"06:51.605","Text":"this is 2 times 5^3 minus 2^3 times 5^2."},{"Start":"06:51.605 ","End":"06:57.515","Text":"Let\u0027s see, 5^3 is 125,"},{"Start":"06:57.515 ","End":"07:04.660","Text":"125 times 2 is 250.2^3 is 8."},{"Start":"07:04.660 ","End":"07:06.425","Text":"Maybe I should\u0027ve written these in small."},{"Start":"07:06.425 ","End":"07:12.020","Text":"This is 125, this is 8, this is 25."},{"Start":"07:12.020 ","End":"07:21.060","Text":"8 times 25 is 200 and 250 minus 200 is 50."},{"Start":"07:21.060 ","End":"07:25.855","Text":"That is exactly what is written on the right-hand side here."},{"Start":"07:25.855 ","End":"07:27.635","Text":"It is correct."},{"Start":"07:27.635 ","End":"07:34.265","Text":"In other words, x=1 is a verified solution and it now really deserves highlighting."},{"Start":"07:34.265 ","End":"07:37.140","Text":"We\u0027re done with part b."}],"ID":8131}],"Thumbnail":null,"ID":56156},{"Name":"(Exponential Equations (Substitution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"7m 57s","ChapterTopicVideoID":8048,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8048.jpeg","UploadDate":"2020-09-30T14:15:21.3800000","DurationForVideoObject":"PT7M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.640","Text":"Here again we have an exercise which is a couple of exercises,"},{"Start":"00:05.640 ","End":"00:08.385","Text":"each of them being an exponential equation."},{"Start":"00:08.385 ","End":"00:11.010","Text":"This time we have a bit of a twist."},{"Start":"00:11.010 ","End":"00:12.360","Text":"In fact, in both of them,"},{"Start":"00:12.360 ","End":"00:15.300","Text":"we have something which is really a hybrid"},{"Start":"00:15.300 ","End":"00:19.365","Text":"between an exponential equation and a quadratic equation."},{"Start":"00:19.365 ","End":"00:22.905","Text":"Let me show you how we go about solving this,"},{"Start":"00:22.905 ","End":"00:27.165","Text":"and let\u0027s begin with part A and we have the formulas as needed."},{"Start":"00:27.165 ","End":"00:31.905","Text":"The first thing I\u0027m going to do is on this 2^2x,"},{"Start":"00:31.905 ","End":"00:35.445","Text":"I\u0027m going to use this formula."},{"Start":"00:35.445 ","End":"00:38.415","Text":"This seems to be the 1 where you use most frequently,"},{"Start":"00:38.415 ","End":"00:44.840","Text":"and 2^2x will be (2^x)2."},{"Start":"00:44.840 ","End":"00:46.465","Text":"A word of an explanation."},{"Start":"00:46.465 ","End":"00:50.195","Text":"If I just did it according to the order 2^2x,"},{"Start":"00:50.195 ","End":"00:54.510","Text":"I\u0027d get (2^2)x,"},{"Start":"00:54.510 ","End":"00:55.790","Text":"but I don\u0027t want that."},{"Start":"00:55.790 ","End":"01:01.070","Text":"I can consider 2^2x to be also 2^x times 2."},{"Start":"01:01.070 ","End":"01:03.740","Text":"There\u0027s no reason I can\u0027t take the x first and then the 2."},{"Start":"01:03.740 ","End":"01:09.400","Text":"Then if I just go according to the formula, it\u0027s (2^x)2."},{"Start":"01:09.400 ","End":"01:11.250","Text":"This is what suits me better,"},{"Start":"01:11.250 ","End":"01:13.140","Text":"so this is the one I\u0027m going to use."},{"Start":"01:13.140 ","End":"01:18.980","Text":"Here, I\u0027ll write this as (2^x)2, but of course,"},{"Start":"01:18.980 ","End":"01:20.285","Text":"there\u0027s still a 2 here,"},{"Start":"01:20.285 ","End":"01:22.340","Text":"and the rest of it just as is,"},{"Start":"01:22.340 ","End":"01:28.055","Text":"9 times 2^x plus 4 equals 0."},{"Start":"01:28.055 ","End":"01:31.760","Text":"Now, what did I mean by saying quadratic equation?"},{"Start":"01:31.760 ","End":"01:37.265","Text":"Well, it turns out the way to solve this is by a method called substitution."},{"Start":"01:37.265 ","End":"01:39.365","Text":"Instead of 2^x,"},{"Start":"01:39.365 ","End":"01:41.455","Text":"I\u0027ll call it by some other letter,"},{"Start":"01:41.455 ","End":"01:42.880","Text":"and what letter am I going to choose?"},{"Start":"01:42.880 ","End":"01:44.420","Text":"Well, choose y."},{"Start":"01:44.420 ","End":"01:46.490","Text":"If I do the substitution,"},{"Start":"01:46.490 ","End":"01:51.395","Text":"I\u0027ll just write down that y is going to be the same as 2^x,"},{"Start":"01:51.395 ","End":"01:55.185","Text":"and then this equation becomes 2,"},{"Start":"01:55.185 ","End":"02:05.655","Text":"and then 2^x is y^2 minus 9 and 2^x is y plus 4=0."},{"Start":"02:05.655 ","End":"02:08.015","Text":"Straightforward quadratic equation."},{"Start":"02:08.015 ","End":"02:09.800","Text":"We\u0027ll solve for y and at the end,"},{"Start":"02:09.800 ","End":"02:12.050","Text":"we\u0027ll see how we get back to x."},{"Start":"02:12.050 ","End":"02:16.490","Text":"Using the formula, y is equal to minus"},{"Start":"02:16.490 ","End":"02:23.870","Text":"b9 plus or minus the square root of b^2 minus 9^2."},{"Start":"02:23.870 ","End":"02:29.435","Text":"I can write it as 9^2 minus 4ac minus 4 times"},{"Start":"02:29.435 ","End":"02:35.685","Text":"2 times 4 over 2a is 2 times 2."},{"Start":"02:35.685 ","End":"02:39.390","Text":"I need a bit more room. There we go."},{"Start":"02:39.390 ","End":"02:47.705","Text":"Continuing, this is equal to 9 plus or minus the square root of, let\u0027s see,"},{"Start":"02:47.705 ","End":"02:52.365","Text":"4 times 2 times 4 is 32,"},{"Start":"02:52.365 ","End":"02:53.430","Text":"this is 81,"},{"Start":"02:53.430 ","End":"02:56.040","Text":"81 minus 32,"},{"Start":"02:56.040 ","End":"03:01.605","Text":"I make it 49 over 4."},{"Start":"03:01.605 ","End":"03:04.010","Text":"Well, square root of 49,"},{"Start":"03:04.010 ","End":"03:05.255","Text":"we know what that is."},{"Start":"03:05.255 ","End":"03:06.530","Text":"This is 7,"},{"Start":"03:06.530 ","End":"03:09.131","Text":"so I\u0027ll save a step here,"},{"Start":"03:09.131 ","End":"03:14.865","Text":"so by taking the plus is 9 plus 7 over 4,"},{"Start":"03:14.865 ","End":"03:17.640","Text":"16 over 4 is 4."},{"Start":"03:17.640 ","End":"03:19.385","Text":"If we take the minus,"},{"Start":"03:19.385 ","End":"03:23.060","Text":"then we\u0027ve got 9 minus 7 over 4."},{"Start":"03:23.060 ","End":"03:28.085","Text":"That\u0027s 2 over 4, that is 1/2."},{"Start":"03:28.085 ","End":"03:29.900","Text":"Now, these are the solutions,"},{"Start":"03:29.900 ","End":"03:31.310","Text":"but not for x,"},{"Start":"03:31.310 ","End":"03:33.635","Text":"these are the solutions for y."},{"Start":"03:33.635 ","End":"03:36.230","Text":"We still have to find x,"},{"Start":"03:36.230 ","End":"03:38.040","Text":"but y is 2^x,"},{"Start":"03:38.040 ","End":"03:41.810","Text":"so what do I get from here is that instead of equaling y,"},{"Start":"03:41.810 ","End":"03:45.905","Text":"I can write that this is equal to 2^x,"},{"Start":"03:45.905 ","End":"03:51.470","Text":"or better still, let me just keep it separate and write 2^x=4."},{"Start":"03:51.470 ","End":"03:53.900","Text":"But if 2^x is 4,"},{"Start":"03:53.900 ","End":"03:56.000","Text":"4 is 2^2,"},{"Start":"03:56.000 ","End":"04:00.545","Text":"and that will use our technique with same base, compare the exponents."},{"Start":"04:00.545 ","End":"04:06.615","Text":"From here, we get that x=2."},{"Start":"04:06.615 ","End":"04:12.165","Text":"For this one, we get also y is 2^x."},{"Start":"04:12.165 ","End":"04:13.845","Text":"This is the answer for y,"},{"Start":"04:13.845 ","End":"04:16.335","Text":"so 2^x is 1/2,"},{"Start":"04:16.335 ","End":"04:19.950","Text":"and 1/2 we\u0027ve seen this before is 2^-1,"},{"Start":"04:19.950 ","End":"04:24.395","Text":"and so comparing the exponents in the first and last bits,"},{"Start":"04:24.395 ","End":"04:29.075","Text":"we\u0027ve got from here that x equals minus 1."},{"Start":"04:29.075 ","End":"04:32.630","Text":"These are the answers for x and that\u0027s what we\u0027re looking for,"},{"Start":"04:32.630 ","End":"04:35.795","Text":"y was just a stepping stone, an intermediary."},{"Start":"04:35.795 ","End":"04:37.940","Text":"We\u0027re done with part A."},{"Start":"04:37.940 ","End":"04:40.180","Text":"Let\u0027s go on to part B."},{"Start":"04:40.180 ","End":"04:45.080","Text":"Let\u0027s use the same trick here, just slightly different."},{"Start":"04:45.080 ","End":"04:49.520","Text":"Notice that here I have 5^x and here I have 25^x,"},{"Start":"04:49.520 ","End":"04:53.905","Text":"what I would really like is 5^x2 so I can work like before."},{"Start":"04:53.905 ","End":"04:56.310","Text":"But notice that 25 is 5^2,"},{"Start":"04:56.310 ","End":"05:03.755","Text":"so let me just manipulate this first term here at the side, 25^x is 5^2."},{"Start":"05:03.755 ","End":"05:08.360","Text":"That\u0027s 25^x, which is 5^2x."},{"Start":"05:08.360 ","End":"05:09.920","Text":"Now in the previous exercise,"},{"Start":"05:09.920 ","End":"05:15.560","Text":"we said that it\u0027s better for us if we write it as first the x and then the 2,"},{"Start":"05:15.560 ","End":"05:20.585","Text":"then this equation is more useful to us because if this is 2,"},{"Start":"05:20.585 ","End":"05:26.585","Text":"then we have this as (5^x)2."},{"Start":"05:26.585 ","End":"05:28.415","Text":"Putting that back here,"},{"Start":"05:28.415 ","End":"05:38.750","Text":"we have (5^x)2 minus 6 times 5^x plus 5=0."},{"Start":"05:38.750 ","End":"05:40.565","Text":"The same trick as before,"},{"Start":"05:40.565 ","End":"05:42.920","Text":"we\u0027re going to make a substitution."},{"Start":"05:42.920 ","End":"05:47.675","Text":"We\u0027ll call it y as 5^x,"},{"Start":"05:47.675 ","End":"05:50.840","Text":"and then wherever where we see 5^x, we put y,"},{"Start":"05:50.840 ","End":"06:00.000","Text":"so we get y^2 minus 6 times y plus 5=0."},{"Start":"06:00.000 ","End":"06:04.250","Text":"Once again, we have a quadratic equation but in y,"},{"Start":"06:04.250 ","End":"06:06.469","Text":"so using the formula,"},{"Start":"06:06.469 ","End":"06:08.165","Text":"I\u0027ll continue over here,"},{"Start":"06:08.165 ","End":"06:18.275","Text":"we\u0027ve got the y equals minus b plus or minus the square root of b^2."},{"Start":"06:18.275 ","End":"06:22.894","Text":"Let\u0027s write it already, minus 6 times minus 6 is 36 minus"},{"Start":"06:22.894 ","End":"06:29.220","Text":"4ac is 4 times 1 times 5 is 20 over 2a,"},{"Start":"06:29.220 ","End":"06:30.330","Text":"a is 1,"},{"Start":"06:30.330 ","End":"06:31.935","Text":"so 2a is 2,"},{"Start":"06:31.935 ","End":"06:39.140","Text":"so what we have is 6 plus or minus 36 minus 20 is 16."},{"Start":"06:39.140 ","End":"06:42.550","Text":"Square root of 16 is 4 over 2."},{"Start":"06:42.550 ","End":"06:45.050","Text":"Taking the plus and the minus separately,"},{"Start":"06:45.050 ","End":"06:48.335","Text":"we take the plus 6 plus 10 over 2 is 5."},{"Start":"06:48.335 ","End":"06:51.295","Text":"6 minus 4 over 2 is 2 over 2 is 1,"},{"Start":"06:51.295 ","End":"06:54.850","Text":"and remember these are not the solutions for x,"},{"Start":"06:54.850 ","End":"06:57.185","Text":"these are the solutions for y,"},{"Start":"06:57.185 ","End":"07:01.459","Text":"but y is 5^x,"},{"Start":"07:01.459 ","End":"07:04.650","Text":"so either 5^x=5,"},{"Start":"07:04.650 ","End":"07:09.360","Text":"and then we\u0027ll see what this means or 5^x=1,"},{"Start":"07:09.360 ","End":"07:11.040","Text":"and we\u0027ll see what that means."},{"Start":"07:11.040 ","End":"07:13.770","Text":"Here, we have 5^x is 5."},{"Start":"07:13.770 ","End":"07:17.190","Text":"That means that 5^x=5^1."},{"Start":"07:17.190 ","End":"07:21.280","Text":"I\u0027m deliberately writing it 5^1 so I can see I\u0027ve got the same base."},{"Start":"07:21.280 ","End":"07:28.780","Text":"Compare the exponents, x=1 and if 5^x=1 then 5^x,"},{"Start":"07:28.780 ","End":"07:31.990","Text":"I\u0027m going to write 1 as 5^0."},{"Start":"07:31.990 ","End":"07:37.400","Text":"This is the first time we\u0027re using this formula which says that anything to the 0 is 1,"},{"Start":"07:37.400 ","End":"07:39.080","Text":"and so once again,"},{"Start":"07:39.080 ","End":"07:42.530","Text":"same base, compare the exponents x=0,"},{"Start":"07:42.530 ","End":"07:47.635","Text":"and these are the solutions for x, not y."},{"Start":"07:47.635 ","End":"07:50.090","Text":"You can get confused and stop here."},{"Start":"07:50.090 ","End":"07:53.090","Text":"Remember this is the intermediate step is y."},{"Start":"07:53.090 ","End":"07:57.300","Text":"We\u0027re done with part B and that\u0027s it."}],"ID":8132},{"Watched":false,"Name":"Exercise 2","Duration":"9m 28s","ChapterTopicVideoID":8039,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8039.jpeg","UploadDate":"2020-09-30T14:19:17.5600000","DurationForVideoObject":"PT9M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.010","Text":"Here we have a couple of equations to solve they\u0027re each exponential."},{"Start":"00:05.010 ","End":"00:07.770","Text":"I can tell you what they both have in common,"},{"Start":"00:07.770 ","End":"00:10.500","Text":"is that in both cases we\u0027re going to use a substitution."},{"Start":"00:10.500 ","End":"00:13.215","Text":"In the first case, 3^x,"},{"Start":"00:13.215 ","End":"00:16.545","Text":"and then the second case, it\u0027ll be 6^x."},{"Start":"00:16.545 ","End":"00:19.020","Text":"Let\u0027s see, we\u0027ll start with Part A and as"},{"Start":"00:19.020 ","End":"00:21.630","Text":"usual we have our formulas in case we need them."},{"Start":"00:21.630 ","End":"00:25.935","Text":"I want to see if I can get everything in terms of 3^x,"},{"Start":"00:25.935 ","End":"00:29.355","Text":"and then I\u0027ll have a chance because 81 is 3^4."},{"Start":"00:29.355 ","End":"00:35.820","Text":"Let\u0027s start off by writing 3 times 3^4 times,"},{"Start":"00:35.820 ","End":"00:45.155","Text":"I\u0027d rather write this as 1/2x plus 5 times 3^x minus 2=0."},{"Start":"00:45.155 ","End":"00:49.879","Text":"Now let\u0027s look at this thing here that\u0027s worked on the side."},{"Start":"00:49.879 ","End":"00:55.165","Text":"I have 3^4^1/2x and"},{"Start":"00:55.165 ","End":"00:59.540","Text":"I\u0027m looking at this formula and I\u0027m going to read it from right to left."},{"Start":"00:59.540 ","End":"01:02.770","Text":"A power of a power, multiply the powers."},{"Start":"01:02.770 ","End":"01:06.650","Text":"I can do that here and what I get is multiply the exponents."},{"Start":"01:06.650 ","End":"01:11.075","Text":"I get 4 times 1/2 is 2, this is 3^2x."},{"Start":"01:11.075 ","End":"01:16.565","Text":"Now 2x is the same as x times 2."},{"Start":"01:16.565 ","End":"01:18.125","Text":"It doesn\u0027t make any difference."},{"Start":"01:18.125 ","End":"01:22.550","Text":"But if I want to use this formula again this time in this direction,"},{"Start":"01:22.550 ","End":"01:27.260","Text":"I can write that this is equal to 3^x^2."},{"Start":"01:27.260 ","End":"01:30.230","Text":"It\u0027s a standard trick to reverse the order."},{"Start":"01:30.230 ","End":"01:33.695","Text":"Now I can really write everything in terms of 3^x."},{"Start":"01:33.695 ","End":"01:37.190","Text":"I get 3 times 3^x^2,"},{"Start":"01:37.190 ","End":"01:39.245","Text":"that\u0027s from the side exercise,"},{"Start":"01:39.245 ","End":"01:44.360","Text":"plus 5 times 3^x minus 2=0."},{"Start":"01:44.360 ","End":"01:45.980","Text":"Now comes the substitution."},{"Start":"01:45.980 ","End":"01:48.125","Text":"Everything is in terms of 3^x,"},{"Start":"01:48.125 ","End":"01:51.110","Text":"I\u0027ll let that equal y. I\u0027ll write that down."},{"Start":"01:51.110 ","End":"01:53.030","Text":"Y equals 3^x,"},{"Start":"01:53.030 ","End":"01:55.565","Text":"and now we get 3 times,"},{"Start":"01:55.565 ","End":"02:02.610","Text":"this is y^2 plus 5y minus 2 equals 0."},{"Start":"02:02.610 ","End":"02:07.655","Text":"That\u0027s a nice quadratic equation in y, so let\u0027s see."},{"Start":"02:07.655 ","End":"02:09.080","Text":"If we use the formula,"},{"Start":"02:09.080 ","End":"02:15.260","Text":"we get y equals minus b plus or minus the square root"},{"Start":"02:15.260 ","End":"02:22.880","Text":"of b^2 minus 4ac/2a."},{"Start":"02:22.880 ","End":"02:24.740","Text":"Let\u0027s see what that gives."},{"Start":"02:24.740 ","End":"02:28.655","Text":"The first thing I want to do is see what\u0027s under the square root sign."},{"Start":"02:28.655 ","End":"02:30.635","Text":"I\u0027ll do that at the side here,"},{"Start":"02:30.635 ","End":"02:36.950","Text":"5^2 is 25 and this becomes a plus because it\u0027s minus,"},{"Start":"02:36.950 ","End":"02:42.650","Text":"minus 4 times 3 times 2 is 24,"},{"Start":"02:42.650 ","End":"02:45.170","Text":"that gives me 49."},{"Start":"02:45.170 ","End":"02:48.770","Text":"I know that the square root of 49 is 7,"},{"Start":"02:48.770 ","End":"02:51.060","Text":"7 times 7 is 49."},{"Start":"02:51.060 ","End":"02:59.370","Text":"We get here, minus 5 plus or minus 7/6."},{"Start":"02:59.370 ","End":"03:02.330","Text":"Now, this should branch into 2 possibilities,"},{"Start":"03:02.330 ","End":"03:04.595","Text":"the plus and the minus."},{"Start":"03:04.595 ","End":"03:06.185","Text":"If I take the plus,"},{"Start":"03:06.185 ","End":"03:11.780","Text":"that\u0027s minus 5 plus 7 is 2, 2/6 is 1/3."},{"Start":"03:11.780 ","End":"03:13.775","Text":"If I take the minus,"},{"Start":"03:13.775 ","End":"03:15.710","Text":"minus 5 minus 7,"},{"Start":"03:15.710 ","End":"03:17.600","Text":"which is minus 12/6,"},{"Start":"03:17.600 ","End":"03:19.780","Text":"I get minus 2."},{"Start":"03:19.780 ","End":"03:22.560","Text":"These are the 2 solutions I get for y,"},{"Start":"03:22.560 ","End":"03:24.809","Text":"but what I want is x."},{"Start":"03:24.809 ","End":"03:26.715","Text":"That gives me 2 possibilities."},{"Start":"03:26.715 ","End":"03:32.460","Text":"Either 3^x equals 1/3,"},{"Start":"03:32.460 ","End":"03:34.980","Text":"or 3^x equals minus 2."},{"Start":"03:34.980 ","End":"03:40.235","Text":"The first case, I solve it by writing 1/3 as 3^-1."},{"Start":"03:40.235 ","End":"03:44.150","Text":"It\u0027s something we\u0027ve seen often a reciprocal is like a power of minus 1,"},{"Start":"03:44.150 ","End":"03:49.165","Text":"and it\u0027s just a straightforward use of the first formula with n=1."},{"Start":"03:49.165 ","End":"03:53.445","Text":"Now we have same base 3 and 3,"},{"Start":"03:53.445 ","End":"03:59.825","Text":"so we can compare exponents so this gives us that x equals minus 1 right away."},{"Start":"03:59.825 ","End":"04:01.805","Text":"Now if we try to solve this one,"},{"Start":"04:01.805 ","End":"04:03.740","Text":"I say this one\u0027s impossible."},{"Start":"04:03.740 ","End":"04:07.325","Text":"A positive number to any power is always going to be positive."},{"Start":"04:07.325 ","End":"04:10.340","Text":"Positive to the power of positive is positive."},{"Start":"04:10.340 ","End":"04:12.170","Text":"If I take it to a negative power,"},{"Start":"04:12.170 ","End":"04:14.960","Text":"it\u0027s like 1 over a positive number and still positive,"},{"Start":"04:14.960 ","End":"04:19.260","Text":"it\u0027s never going to be negative so this has to be ruled out,"},{"Start":"04:19.260 ","End":"04:24.055","Text":"and so the only valid solution is x equals minus 1,"},{"Start":"04:24.055 ","End":"04:26.425","Text":"and that\u0027s done for Part A."},{"Start":"04:26.425 ","End":"04:31.820","Text":"Let\u0027s move on to Part B. I\u0027ll go get that formula sheet,"},{"Start":"04:31.820 ","End":"04:37.670","Text":"I\u0027m going to use the same idea in Part B and try and get everything in terms of 6^x."},{"Start":"04:37.670 ","End":"04:41.540","Text":"I already see that 36 is 6^2 so I know I have a chance."},{"Start":"04:41.540 ","End":"04:46.130","Text":"Notice that here I have x plus 0.5 here, x plus 1."},{"Start":"04:46.130 ","End":"04:49.270","Text":"So it looks like this formula will come in useful,"},{"Start":"04:49.270 ","End":"04:53.390","Text":"and also I know that this always comes in useful."},{"Start":"04:53.390 ","End":"04:55.088","Text":"I\u0027m going to highlight it already."},{"Start":"04:55.088 ","End":"04:57.580","Text":"Since I see a power of 0 might as well,"},{"Start":"04:57.580 ","End":"05:00.380","Text":"highlight this while I\u0027m highlighting."},{"Start":"05:00.380 ","End":"05:07.040","Text":"Idea is again to get everything in terms of 6 or 6^x to be specific and we"},{"Start":"05:07.040 ","End":"05:14.280","Text":"start off by saying 36 is 6^2 so I have 6^2^x plus,"},{"Start":"05:14.280 ","End":"05:17.790","Text":"let me write the 0.5 as 1/2 minus,"},{"Start":"05:17.790 ","End":"05:20.730","Text":"now here I want to use this formula,"},{"Start":"05:20.730 ","End":"05:27.260","Text":"x plus 1 so it\u0027s 6^x times 6^1,"},{"Start":"05:27.260 ","End":"05:29.840","Text":"the plus becomes a multiplication here."},{"Start":"05:29.840 ","End":"05:31.700","Text":"On the right-hand side, again,"},{"Start":"05:31.700 ","End":"05:40.090","Text":"6^2^1/2x, and here 6^0 we said is 1."},{"Start":"05:40.090 ","End":"05:44.280","Text":"Let\u0027s expand here I mean simplify."},{"Start":"05:44.280 ","End":"05:47.840","Text":"I\u0027m going to use this formula that says when we have an exponent of exponent,"},{"Start":"05:47.840 ","End":"05:49.835","Text":"we multiply the exponents."},{"Start":"05:49.835 ","End":"05:53.450","Text":"So here I have this times this,"},{"Start":"05:53.450 ","End":"05:54.800","Text":"we can do the whole thing at once."},{"Start":"05:54.800 ","End":"05:56.695","Text":"It\u0027s 2x plus 1,"},{"Start":"05:56.695 ","End":"06:00.075","Text":"2 times brackets (x +1/2) is 2x plus 1."},{"Start":"06:00.075 ","End":"06:05.250","Text":"Here I have the 6 times 6^x."},{"Start":"06:05.250 ","End":"06:08.055","Text":"Here I have 2 times 1/2x,"},{"Start":"06:08.055 ","End":"06:13.615","Text":"again using this formula is 6^x and minus 1."},{"Start":"06:13.615 ","End":"06:17.225","Text":"The only thing that isn\u0027t in terms of 6^x is this."},{"Start":"06:17.225 ","End":"06:19.505","Text":"Let me just work on this a bit."},{"Start":"06:19.505 ","End":"06:23.030","Text":"This I can modify with 2 formulas."},{"Start":"06:23.030 ","End":"06:27.110","Text":"First of all, I\u0027m going to use this formula for the plus and separate"},{"Start":"06:27.110 ","End":"06:32.735","Text":"6^2x times 6^1, then 6^2^x."},{"Start":"06:32.735 ","End":"06:34.640","Text":"I think we\u0027ve talked about this trick before."},{"Start":"06:34.640 ","End":"06:37.610","Text":"It\u0027s the same as 6^x times 2,"},{"Start":"06:37.610 ","End":"06:44.060","Text":"which means it is 6^x^2."},{"Start":"06:44.060 ","End":"06:52.045","Text":"Putting that here, this is equal to 6 times 6^x^2."},{"Start":"06:52.045 ","End":"06:55.080","Text":"Just kept this so if I rewrite this,"},{"Start":"06:55.080 ","End":"07:00.540","Text":"I get 6 times 6^x^2"},{"Start":"07:00.540 ","End":"07:06.355","Text":"minus 6 times 6^x equals 6^x minus 1."},{"Start":"07:06.355 ","End":"07:10.005","Text":"At this point everything is in terms of 6^x,"},{"Start":"07:10.005 ","End":"07:14.280","Text":"so this is where we make our substitution y equals 6^x,"},{"Start":"07:14.280 ","End":"07:23.880","Text":"and then we get 6y^2 minus 6y equals y minus 1."},{"Start":"07:23.880 ","End":"07:26.460","Text":"Let\u0027s bring everything over to the left,"},{"Start":"07:26.460 ","End":"07:31.290","Text":"6y^2 minus 6y minus y is minus 7y,"},{"Start":"07:31.290 ","End":"07:36.425","Text":"and plus 1 equals 0, its formula time."},{"Start":"07:36.425 ","End":"07:42.290","Text":"So we get that y equals minus b plus or"},{"Start":"07:42.290 ","End":"07:49.820","Text":"minus the square root of b^2 minus 4ac/2a."},{"Start":"07:49.820 ","End":"07:53.060","Text":"Let\u0027s see what\u0027s under the square root sign,"},{"Start":"07:53.060 ","End":"07:56.945","Text":"4 times 6 is 24,"},{"Start":"07:56.945 ","End":"08:00.565","Text":"49 minus 24 is 25,"},{"Start":"08:00.565 ","End":"08:06.635","Text":"and I know that the square root of 25 is 5,"},{"Start":"08:06.635 ","End":"08:13.340","Text":"so this becomes 7 plus or minus 5 over,"},{"Start":"08:13.340 ","End":"08:15.140","Text":"2 times 6, 12."},{"Start":"08:15.140 ","End":"08:17.945","Text":"Let\u0027s see what possibilities we have for y,"},{"Start":"08:17.945 ","End":"08:22.580","Text":"7 plus 5 is 12 over 12 is 1,"},{"Start":"08:22.580 ","End":"08:28.900","Text":"7 minus 5 is 2/12 is 1/6,"},{"Start":"08:28.900 ","End":"08:31.770","Text":"notice that both are positive."},{"Start":"08:31.770 ","End":"08:35.570","Text":"Unlike in the previous case, in the previous case,"},{"Start":"08:35.570 ","End":"08:39.410","Text":"we had 1 positive and 1 negative and we had to rule it out but here we don\u0027t."},{"Start":"08:39.410 ","End":"08:42.890","Text":"We still remember these are why the way to get back to x is"},{"Start":"08:42.890 ","End":"08:46.340","Text":"through this formula y is 6^x so we have"},{"Start":"08:46.340 ","End":"08:54.050","Text":"that 6^x is equal to 1 or 6^x equals 1/6."},{"Start":"08:54.050 ","End":"08:58.530","Text":"I want to write each of these as 6 to the power of something, 1 is 6^0."},{"Start":"08:58.530 ","End":"09:08.335","Text":"We\u0027ve already done this and 1/6 is 6 to the power of minus 1 using this formula with n=1."},{"Start":"09:08.335 ","End":"09:10.130","Text":"Now we have 2 things."},{"Start":"09:10.130 ","End":"09:17.225","Text":"This term and this term have the same base 6 so I can compare the exponents and get x=0."},{"Start":"09:17.225 ","End":"09:19.385","Text":"Here again, same base 6,"},{"Start":"09:19.385 ","End":"09:22.815","Text":"so x equals minus 1,"},{"Start":"09:22.815 ","End":"09:29.410","Text":"and these are the 2 possibilities for x in Part B and we are done."}],"ID":8133},{"Watched":false,"Name":"Exercise 3","Duration":"12m 42s","ChapterTopicVideoID":8040,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8040.jpeg","UploadDate":"2020-09-30T14:25:05.4570000","DurationForVideoObject":"PT12M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this exercise, we have a couple of equations."},{"Start":"00:03.300 ","End":"00:05.175","Text":"They\u0027re both exponential."},{"Start":"00:05.175 ","End":"00:08.970","Text":"Let\u0027s see what we can do with the first one."},{"Start":"00:08.970 ","End":"00:13.170","Text":"As usual, the formulas are on hand in case we need them."},{"Start":"00:13.170 ","End":"00:17.955","Text":"As I look at it, I see a 10 and I see 100 and I know 100 is 10^2."},{"Start":"00:17.955 ","End":"00:20.760","Text":"Let\u0027s see what we can do with this."},{"Start":"00:20.760 ","End":"00:24.045","Text":"I suggest writing 100 as 10^2 right off."},{"Start":"00:24.045 ","End":"00:31.290","Text":"We have 10^2 to the power of minus x plus 0.5."},{"Start":"00:31.290 ","End":"00:36.030","Text":"Here, I have in the denominator 10^x."},{"Start":"00:36.030 ","End":"00:37.915","Text":"If I look at this formula,"},{"Start":"00:37.915 ","End":"00:42.470","Text":"and I see I can rewrite it as 9 and instead of 1/10 to the x,"},{"Start":"00:42.470 ","End":"00:45.500","Text":"I can have 10 to the minus x."},{"Start":"00:45.500 ","End":"00:47.965","Text":"Just for clarity, I\u0027ll put this in brackets,"},{"Start":"00:47.965 ","End":"00:51.010","Text":"and this equals 1."},{"Start":"00:51.010 ","End":"00:53.720","Text":"Now, I\u0027m going to use this formula,"},{"Start":"00:53.720 ","End":"00:56.945","Text":"but from right to left on this thing here."},{"Start":"00:56.945 ","End":"01:02.030","Text":"What I have to do is multiply the 2 by the minus x plus 0.5."},{"Start":"01:02.030 ","End":"01:04.640","Text":"This is the product of the two exponents."},{"Start":"01:04.640 ","End":"01:06.530","Text":"I get 10 to the power of,"},{"Start":"01:06.530 ","End":"01:08.815","Text":"let\u0027s just multiply it out."},{"Start":"01:08.815 ","End":"01:13.380","Text":"We get minus 2x plus 1,"},{"Start":"01:13.380 ","End":"01:15.690","Text":"because 2 times 0.5 is 1,"},{"Start":"01:15.690 ","End":"01:20.760","Text":"and then plus 9 times 10 to the minus x."},{"Start":"01:20.760 ","End":"01:23.810","Text":"Let\u0027s throw the 1 to the left-hand side because I have the feeling,"},{"Start":"01:23.810 ","End":"01:26.108","Text":"we\u0027re going to get a quadratic equation,"},{"Start":"01:26.108 ","End":"01:28.975","Text":"minus 1 equals 0."},{"Start":"01:28.975 ","End":"01:32.390","Text":"Now, I notice that here we have a 10 to the minus x,"},{"Start":"01:32.390 ","End":"01:33.680","Text":"and if I square that,"},{"Start":"01:33.680 ","End":"01:35.660","Text":"that\u0027s 10 to the minus 2x."},{"Start":"01:35.660 ","End":"01:38.855","Text":"I\u0027d like to get rid of that plus 1 somehow,"},{"Start":"01:38.855 ","End":"01:43.180","Text":"and this is where I use this formula here,"},{"Start":"01:43.180 ","End":"01:46.434","Text":"with m being the minus 2x and n being the 1."},{"Start":"01:46.434 ","End":"01:54.000","Text":"Until I get 10^minus 2x times 10^1,"},{"Start":"01:54.000 ","End":"01:55.620","Text":"which is just 10,"},{"Start":"01:55.620 ","End":"01:59.940","Text":"plus 9 times 10^x,"},{"Start":"01:59.940 ","End":"02:02.810","Text":"minus 1 equals 0."},{"Start":"02:02.810 ","End":"02:05.660","Text":"This is an almost familiar situation."},{"Start":"02:05.660 ","End":"02:10.310","Text":"We\u0027ve had a substitution before with something to the power of x."},{"Start":"02:10.310 ","End":"02:12.635","Text":"This time it\u0027s something to the power of minus x."},{"Start":"02:12.635 ","End":"02:15.385","Text":"No big difference. What I\u0027m going to do,"},{"Start":"02:15.385 ","End":"02:17.595","Text":"let\u0027s get a bit more space here,"},{"Start":"02:17.595 ","End":"02:21.200","Text":"is replace, better with a substitute,"},{"Start":"02:21.200 ","End":"02:25.055","Text":"y equals 10 to the minus x."},{"Start":"02:25.055 ","End":"02:28.370","Text":"Now, 10 to the minus 2x is y^2."},{"Start":"02:28.370 ","End":"02:30.110","Text":"Let me explain again."},{"Start":"02:30.110 ","End":"02:32.210","Text":"10 to the minus 2x,"},{"Start":"02:32.210 ","End":"02:35.660","Text":"I can think of it as minus x times 2."},{"Start":"02:35.660 ","End":"02:40.835","Text":"This will be 10 to the minus x to the power of 2,"},{"Start":"02:40.835 ","End":"02:42.230","Text":"because instead of 2x,"},{"Start":"02:42.230 ","End":"02:43.910","Text":"I can write it as x2,"},{"Start":"02:43.910 ","End":"02:45.694","Text":"and then using this formula,"},{"Start":"02:45.694 ","End":"02:48.330","Text":"and then 10^x is just y."},{"Start":"02:48.330 ","End":"02:51.225","Text":"What I get here is,"},{"Start":"02:51.225 ","End":"02:53.220","Text":"this is y^2 times 10."},{"Start":"02:53.220 ","End":"02:59.269","Text":"Well, better write it as 10 times y^2 plus 9 times y."},{"Start":"02:59.269 ","End":"03:04.170","Text":"This thing is just y minus 1 equals 0."},{"Start":"03:04.170 ","End":"03:06.315","Text":"Now we have a nice quadratic,"},{"Start":"03:06.315 ","End":"03:08.865","Text":"just happens to be in y not in x."},{"Start":"03:08.865 ","End":"03:17.930","Text":"Using the formula, we get that y equals minus b plus or minus the square root"},{"Start":"03:17.930 ","End":"03:27.030","Text":"of b squared is 81 minus 4 times a times c is minus 1,"},{"Start":"03:27.030 ","End":"03:31.595","Text":"and all this over 2a, 2a is 20."},{"Start":"03:31.595 ","End":"03:34.550","Text":"Let\u0027s just look what happens under the square root sign."},{"Start":"03:34.550 ","End":"03:38.600","Text":"I have 81 and this times this times this well is a minus and a minus,"},{"Start":"03:38.600 ","End":"03:39.770","Text":"so it\u0027s plus,"},{"Start":"03:39.770 ","End":"03:42.050","Text":"and 4 times 10 is 40,"},{"Start":"03:42.050 ","End":"03:45.885","Text":"81 plus 40 is 121."},{"Start":"03:45.885 ","End":"03:51.125","Text":"What we have here is a square root of 121, which is 11."},{"Start":"03:51.125 ","End":"03:59.855","Text":"What we get is minus 9 plus or minus 11/20."},{"Start":"03:59.855 ","End":"04:01.699","Text":"If we take the plus,"},{"Start":"04:01.699 ","End":"04:05.910","Text":"we have minus 9 plus 11 is 2/20 is 1/10,"},{"Start":"04:05.910 ","End":"04:08.505","Text":"and if we take the minus,"},{"Start":"04:08.505 ","End":"04:10.915","Text":"we get minus 9,"},{"Start":"04:10.915 ","End":"04:14.795","Text":"minus 11 is minus 20,"},{"Start":"04:14.795 ","End":"04:18.950","Text":"minus 20/20 is minus 1."},{"Start":"04:18.950 ","End":"04:21.338","Text":"Now, this is not the answer for x,"},{"Start":"04:21.338 ","End":"04:23.105","Text":"this is the answer for y."},{"Start":"04:23.105 ","End":"04:26.750","Text":"It\u0027s y that can be 1/10 or minus 1."},{"Start":"04:26.750 ","End":"04:30.110","Text":"Now, we have to remember what y is so we can get back to x."},{"Start":"04:30.110 ","End":"04:32.045","Text":"This is 10^x,"},{"Start":"04:32.045 ","End":"04:34.175","Text":"we have two possibilities."},{"Start":"04:34.175 ","End":"04:38.795","Text":"Either 10 to the minus x is 1/10,"},{"Start":"04:38.795 ","End":"04:44.150","Text":"and 1/10 is just 10 to the power of minus 1."},{"Start":"04:44.150 ","End":"04:45.710","Text":"We use this a lot,"},{"Start":"04:45.710 ","End":"04:48.635","Text":"1/10 to the 1 is 10 to the minus 1."},{"Start":"04:48.635 ","End":"04:50.660","Text":"If this is the case,"},{"Start":"04:50.660 ","End":"04:54.510","Text":"then we get by comparing the exponents,"},{"Start":"04:54.510 ","End":"05:01.225","Text":"minus x equals minus 1 or in other words, x equals 1."},{"Start":"05:01.225 ","End":"05:03.425","Text":"If we take the other branch,"},{"Start":"05:03.425 ","End":"05:09.165","Text":"I would like to write this minus 1 as 10 to the something,"},{"Start":"05:09.165 ","End":"05:12.005","Text":"and then compare that something to minus x."},{"Start":"05:12.005 ","End":"05:16.730","Text":"But 10 to the power of something is never negative, it\u0027s always positive."},{"Start":"05:16.730 ","End":"05:18.530","Text":"10 to a positive is positive,"},{"Start":"05:18.530 ","End":"05:22.655","Text":"and 10 to a negative is 1/10 to the positive, it\u0027s also positive."},{"Start":"05:22.655 ","End":"05:26.225","Text":"This one is impossible, not possible."},{"Start":"05:26.225 ","End":"05:30.050","Text":"The only acceptable answer for x is x equals 1,"},{"Start":"05:30.050 ","End":"05:32.875","Text":"and we\u0027re done with Part A."},{"Start":"05:32.875 ","End":"05:35.460","Text":"From time to time, I like to check my solutions,"},{"Start":"05:35.460 ","End":"05:37.310","Text":"so before we move on to Part B,"},{"Start":"05:37.310 ","End":"05:45.205","Text":"let\u0027s see if x=1 really is a solution by substituting in the original equation."},{"Start":"05:45.205 ","End":"05:49.725","Text":"I\u0027m substituting x equals 1 in this equation,"},{"Start":"05:49.725 ","End":"05:51.180","Text":"and see what we get."},{"Start":"05:51.180 ","End":"05:59.160","Text":"We get 100 to the power of minus 1 plus 0.5,"},{"Start":"05:59.160 ","End":"06:05.480","Text":"plus 9/10^1, and I\u0027m just going to"},{"Start":"06:05.480 ","End":"06:07.730","Text":"work on the left-hand side and see if we can reach"},{"Start":"06:07.730 ","End":"06:11.780","Text":"the right-hand side because now we don\u0027t know that it\u0027s equal to 1."},{"Start":"06:11.780 ","End":"06:13.190","Text":"That\u0027s what we\u0027re trying to check."},{"Start":"06:13.190 ","End":"06:15.050","Text":"Let\u0027s see what this is equal to,"},{"Start":"06:15.050 ","End":"06:24.620","Text":"this is equal to 100 to the power of minus 1 plus 0.5 is minus 0.5."},{"Start":"06:24.620 ","End":"06:29.485","Text":"Let me write it as minus 1/2 plus 9/10,"},{"Start":"06:29.485 ","End":"06:31.605","Text":"and this is equal 2."},{"Start":"06:31.605 ","End":"06:35.960","Text":"Now, let\u0027s take care of the minus first using this formula."},{"Start":"06:35.960 ","End":"06:42.020","Text":"This is 1/100^1/2 again,"},{"Start":"06:42.020 ","End":"06:48.600","Text":"plus 9/10, and that will take care of the half using this formula here."},{"Start":"06:48.600 ","End":"06:56.205","Text":"What we get is 1 over the square root of 100 plus 9/10, which equals,"},{"Start":"06:56.205 ","End":"06:57.540","Text":"square root of 100 is 10,"},{"Start":"06:57.540 ","End":"07:05.230","Text":"so it\u0027s 1/10 plus 9/10 and 1/10 plus 9/10 is equal to 1."},{"Start":"07:05.230 ","End":"07:08.375","Text":"We really did get the right-hand side,"},{"Start":"07:08.375 ","End":"07:13.280","Text":"so x equals 1 is now a verified solution for Part A."},{"Start":"07:13.280 ","End":"07:15.120","Text":"Finally, let\u0027s move on to Part B,"},{"Start":"07:15.120 ","End":"07:19.940","Text":"but I\u0027m taking the formula with me, move it down."},{"Start":"07:19.940 ","End":"07:24.110","Text":"Part B, I look at this and I see all numbers."},{"Start":"07:24.110 ","End":"07:25.940","Text":"I see 5/2,"},{"Start":"07:25.940 ","End":"07:30.035","Text":"I see 6 and 1/4, I see 25/4."},{"Start":"07:30.035 ","End":"07:35.000","Text":"Actually, these are all related because let\u0027s do some arithmetic at the side,"},{"Start":"07:35.000 ","End":"07:40.880","Text":"5/2 squared and using this formula here,"},{"Start":"07:40.880 ","End":"07:42.988","Text":"well you know how to square fractions,"},{"Start":"07:42.988 ","End":"07:45.245","Text":"we must square the top and we square the bottom,"},{"Start":"07:45.245 ","End":"07:48.840","Text":"is 5^2/ 2^2,"},{"Start":"07:48.840 ","End":"07:51.045","Text":"5^2 is 25,"},{"Start":"07:51.045 ","End":"07:54.540","Text":"2^2 is 4, so that\u0027s 25/4."},{"Start":"07:54.540 ","End":"07:56.279","Text":"We have these two related,"},{"Start":"07:56.279 ","End":"07:58.185","Text":"that this one is the square of this one."},{"Start":"07:58.185 ","End":"08:01.220","Text":"But 25/4 is an improper fraction."},{"Start":"08:01.220 ","End":"08:04.025","Text":"Let\u0027s see what would happen if we convert it to a mixed number,"},{"Start":"08:04.025 ","End":"08:08.895","Text":"4 into 25 goes 6 times,1 leftover."},{"Start":"08:08.895 ","End":"08:10.685","Text":"This is 6 and 1/4,"},{"Start":"08:10.685 ","End":"08:16.155","Text":"6 and 1/4 and 25/4 are the same and they\u0027re both 5/2 squared."},{"Start":"08:16.155 ","End":"08:20.039","Text":"Looks like 5/2 will be our base."},{"Start":"08:20.039 ","End":"08:23.690","Text":"Let\u0027s write everything in terms of 5/2."},{"Start":"08:23.690 ","End":"08:28.955","Text":"Here, I have 5/2 squared to the power of"},{"Start":"08:28.955 ","End":"08:36.125","Text":"x minus twice 5/2 to the power of x plus 1,"},{"Start":"08:36.125 ","End":"08:42.560","Text":"plus 5/2 squared equals 0."},{"Start":"08:42.560 ","End":"08:46.720","Text":"Let\u0027s see, if I use this formula,"},{"Start":"08:46.720 ","End":"08:48.025","Text":"power of a power,"},{"Start":"08:48.025 ","End":"08:55.575","Text":"multiply the powers, (5/2)^2x minus."},{"Start":"08:55.575 ","End":"09:03.455","Text":"Now here, I\u0027ll make use of the formula here with x plus 1."},{"Start":"09:03.455 ","End":"09:09.105","Text":"This is twice (5/2)^x,"},{"Start":"09:09.105 ","End":"09:19.065","Text":"times (5/2)^1 is just 5/2, plus (5/2)^2=0."},{"Start":"09:19.065 ","End":"09:22.100","Text":"Now, I can make a substitution."},{"Start":"09:22.100 ","End":"09:25.270","Text":"I\u0027ll hold off just for one more line to simplify things."},{"Start":"09:25.270 ","End":"09:30.190","Text":"I see that I can multiply 2 times 5/2,"},{"Start":"09:30.190 ","End":"09:31.810","Text":"and that will give me 5,"},{"Start":"09:31.810 ","End":"09:34.100","Text":"the 2\u0027s will cancel."},{"Start":"09:34.100 ","End":"09:39.945","Text":"This term is minus 5 times (5/2)^x,"},{"Start":"09:39.945 ","End":"09:42.090","Text":"and here we use our usual trick."},{"Start":"09:42.090 ","End":"09:43.590","Text":"Instead of 2 times x,"},{"Start":"09:43.590 ","End":"09:45.510","Text":"we\u0027ll write it as x times 2,"},{"Start":"09:45.510 ","End":"09:51.675","Text":"and then it\u0027s (5/2)^x squared."},{"Start":"09:51.675 ","End":"09:55.790","Text":"I don\u0027t think it really helps me to write this as (5/2)^2,"},{"Start":"09:55.790 ","End":"10:01.910","Text":"so I\u0027ll move it back to 25/4 and equals 0."},{"Start":"10:01.910 ","End":"10:05.255","Text":"Now we\u0027re ready for the substitution to make it a quadratic."},{"Start":"10:05.255 ","End":"10:11.240","Text":"We let y=(5/2)^x,"},{"Start":"10:11.240 ","End":"10:22.195","Text":"and then we get y^2 minus 5y plus 25/4=0."},{"Start":"10:22.195 ","End":"10:27.090","Text":"I prefer to multiply it by 4 and not have fractions."},{"Start":"10:27.090 ","End":"10:28.310","Text":"When I use the formula,"},{"Start":"10:28.310 ","End":"10:30.574","Text":"it\u0027s easier if there\u0027s no fractions."},{"Start":"10:30.574 ","End":"10:32.504","Text":"Let\u0027s multiply this by 4,"},{"Start":"10:32.504 ","End":"10:36.735","Text":"multiply by 4 on both sides,"},{"Start":"10:36.735 ","End":"10:41.735","Text":"and we get, multiply everything by 4, 4y^2."},{"Start":"10:41.735 ","End":"10:44.805","Text":"5 times 4, 20y,"},{"Start":"10:44.805 ","End":"10:49.395","Text":"and 25/4 multiplied by 4 is just 25,"},{"Start":"10:49.395 ","End":"10:52.170","Text":"0 times 4 is 0."},{"Start":"10:52.170 ","End":"10:55.789","Text":"Now I\u0027m going to use the formula to find y."},{"Start":"10:55.789 ","End":"11:01.550","Text":"The formula says we need minus b, that\u0027s 20."},{"Start":"11:01.550 ","End":"11:03.250","Text":"I\u0027ll write that y equals here,"},{"Start":"11:03.250 ","End":"11:04.890","Text":"a minus b is 20,"},{"Start":"11:04.890 ","End":"11:12.915","Text":"plus or minus the square root of b^2 is 400 minus 4ac,"},{"Start":"11:12.915 ","End":"11:17.445","Text":"4 times 4 times 25."},{"Start":"11:17.445 ","End":"11:22.155","Text":"This all over 2a, which is 8."},{"Start":"11:22.155 ","End":"11:26.465","Text":"Let\u0027s see, what interests me is under the square root sign."},{"Start":"11:26.465 ","End":"11:29.435","Text":"Now, 4 times 25 is 100,"},{"Start":"11:29.435 ","End":"11:31.325","Text":"times 4 is 400,"},{"Start":"11:31.325 ","End":"11:35.335","Text":"so what\u0027s under the square root sign is 0."},{"Start":"11:35.335 ","End":"11:40.860","Text":"I get 20 plus or minus 0/8."},{"Start":"11:40.860 ","End":"11:44.915","Text":"Because plus or minus 0 doesn\u0027t change anything it means there\u0027s only one solution."},{"Start":"11:44.915 ","End":"11:46.730","Text":"I don\u0027t have to take the plus or the minus,"},{"Start":"11:46.730 ","End":"11:54.660","Text":"I just take 20/8 and 20/8 is 2 and 1/2 but"},{"Start":"11:54.660 ","End":"11:59.400","Text":"2 and 1/2 is the same as 5/2 and I want"},{"Start":"11:59.400 ","End":"12:04.340","Text":"to write it as 5/2 because we\u0027ve already been working with 5/2."},{"Start":"12:04.340 ","End":"12:10.530","Text":"We found the solution that y equals 5/2 but we\u0027re not looking for y,"},{"Start":"12:10.530 ","End":"12:19.335","Text":"we\u0027re looking for x, (5/2)^x=5/2."},{"Start":"12:19.335 ","End":"12:20.790","Text":"Now, how do I solve this?"},{"Start":"12:20.790 ","End":"12:23.955","Text":"Quite simply, I can rewrite 5/2,"},{"Start":"12:23.955 ","End":"12:28.515","Text":"as (5/2)^1, and anything to the power of 1 is itself."},{"Start":"12:28.515 ","End":"12:30.540","Text":"Now I have the same base,"},{"Start":"12:30.540 ","End":"12:32.795","Text":"5/2 is that base, of course."},{"Start":"12:32.795 ","End":"12:36.260","Text":"Then we compare the exponent and we immediately get"},{"Start":"12:36.260 ","End":"12:42.180","Text":"x=1 and this is our only answer, and we\u0027re done."}],"ID":8134},{"Watched":false,"Name":"Exercise 4","Duration":"11m 18s","ChapterTopicVideoID":8041,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8041.jpeg","UploadDate":"2020-09-30T14:29:50.8830000","DurationForVideoObject":"PT11M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"Yet another exercise with a couple of exponential equations."},{"Start":"00:04.680 ","End":"00:08.610","Text":"As usual, I have the formulas on hand in case we need them."},{"Start":"00:08.610 ","End":"00:10.785","Text":"Let\u0027s start with part a."},{"Start":"00:10.785 ","End":"00:14.460","Text":"Everything looks like base 2, 2^x."},{"Start":"00:14.460 ","End":"00:20.925","Text":"Let\u0027s see if we can bring everything in terms of 2^x and 2^ minus x is also good."},{"Start":"00:20.925 ","End":"00:28.140","Text":"We\u0027re going to use this one and also perhaps this one."},{"Start":"00:28.140 ","End":"00:30.360","Text":"Now, I think we can make do with just this one."},{"Start":"00:30.360 ","End":"00:32.580","Text":"Each time the n is going to be 2."},{"Start":"00:32.580 ","End":"00:34.635","Text":"I have a plus 2 and a plus 2."},{"Start":"00:34.635 ","End":"00:39.920","Text":"This is 2^x times 2^2,"},{"Start":"00:39.920 ","End":"00:46.940","Text":"and here, 2^minus x also times 2^2 equals 10."},{"Start":"00:46.940 ","End":"00:49.550","Text":"What I\u0027m going to do, well,"},{"Start":"00:49.550 ","End":"00:51.860","Text":"first of all, I should really write the 2^2 is 4."},{"Start":"00:51.860 ","End":"00:56.405","Text":"Let\u0027s write this as 4 times 2^x plus,"},{"Start":"00:56.405 ","End":"01:02.595","Text":"again I put the 2^2 in front is 4 times 2^ minus x equals 10."},{"Start":"01:02.595 ","End":"01:07.400","Text":"I suppose we should also cancel because obviously everything is even."},{"Start":"01:07.400 ","End":"01:09.175","Text":"Let\u0027s divide by 2,"},{"Start":"01:09.175 ","End":"01:11.865","Text":"divide both sides by 2."},{"Start":"01:11.865 ","End":"01:16.100","Text":"2 times 2^x plus 2 times."},{"Start":"01:16.100 ","End":"01:19.895","Text":"I\u0027ll leave this for, just for a moment, equals 5."},{"Start":"01:19.895 ","End":"01:22.210","Text":"Now the 2^ minus x,"},{"Start":"01:22.210 ","End":"01:29.150","Text":"I\u0027m going to use this formula and I\u0027m going to write 2^minus x is 1/2^x."},{"Start":"01:29.150 ","End":"01:31.550","Text":"Instead of 2^ minus x,"},{"Start":"01:31.550 ","End":"01:33.829","Text":"1/2^x according to the formula."},{"Start":"01:33.829 ","End":"01:36.590","Text":"Now, I see I have an equation,"},{"Start":"01:36.590 ","End":"01:39.935","Text":"not really in x, but in 2^x."},{"Start":"01:39.935 ","End":"01:43.925","Text":"I mean, I have a 2^x here and a 2^x here."},{"Start":"01:43.925 ","End":"01:46.795","Text":"It looks like I should make a substitution."},{"Start":"01:46.795 ","End":"01:49.710","Text":"Let\u0027s let y equals 2^x."},{"Start":"01:49.710 ","End":"01:53.240","Text":"The whole thing will be a lot simpler and there won\u0027t be any exponents."},{"Start":"01:53.240 ","End":"01:57.140","Text":"We 2y, and I\u0027ll put the 2 on top."},{"Start":"01:57.140 ","End":"02:00.345","Text":"It\u0027s 2/y, it\u0027s 2 times 1/y,"},{"Start":"02:00.345 ","End":"02:03.960","Text":"which is 2/y equals 5."},{"Start":"02:03.960 ","End":"02:06.060","Text":"This looks familiar."},{"Start":"02:06.060 ","End":"02:10.670","Text":"It\u0027s the thing they give you that becomes a quadratic equation."},{"Start":"02:10.670 ","End":"02:12.575","Text":"Quadratic in disguise."},{"Start":"02:12.575 ","End":"02:16.070","Text":"What we have to do is multiply both sides by"},{"Start":"02:16.070 ","End":"02:20.495","Text":"y. I\u0027m going to get rid of fractions by multiplying by y."},{"Start":"02:20.495 ","End":"02:23.090","Text":"But at the end I have to check that I don\u0027t get"},{"Start":"02:23.090 ","End":"02:26.030","Text":"y equals 0 because the denominator can\u0027t be 0."},{"Start":"02:26.030 ","End":"02:30.605","Text":"Let\u0027s just make a note at the end to check that y is not 0."},{"Start":"02:30.605 ","End":"02:37.135","Text":"Multiplying out, well, we can easily do that 2 times y times y is 2 y^2,"},{"Start":"02:37.135 ","End":"02:42.125","Text":"2^y, the y cancels with y that\u0027s plus 2."},{"Start":"02:42.125 ","End":"02:44.335","Text":"On the right we have 5y."},{"Start":"02:44.335 ","End":"02:50.195","Text":"Now a little rearranging will give us a nice quadratic equation."},{"Start":"02:50.195 ","End":"02:54.020","Text":"2y^2, the minus 5y to the other side,"},{"Start":"02:54.020 ","End":"02:56.770","Text":"plus 2 equals 0."},{"Start":"02:56.770 ","End":"02:59.175","Text":"Let\u0027s solve this with the formula."},{"Start":"02:59.175 ","End":"03:05.450","Text":"We have that y is equal to minus b plus or minus"},{"Start":"03:05.450 ","End":"03:13.265","Text":"the square root of b^2 is 25 minus 4ac,"},{"Start":"03:13.265 ","End":"03:18.460","Text":"4 times 2 times 2 all over 2a,"},{"Start":"03:18.460 ","End":"03:20.520","Text":"which is 2 times 2."},{"Start":"03:20.520 ","End":"03:22.530","Text":"Let\u0027s see what that gives us."},{"Start":"03:22.530 ","End":"03:25.860","Text":"4 times 2 times 2 is 16,"},{"Start":"03:25.860 ","End":"03:30.510","Text":"25 minus 16 is 9."},{"Start":"03:30.510 ","End":"03:35.085","Text":"We know that the square root of 9 is 3."},{"Start":"03:35.085 ","End":"03:41.370","Text":"What we get is y equals 5 plus or minus 3,"},{"Start":"03:41.370 ","End":"03:43.380","Text":"2 times 2 is 4."},{"Start":"03:43.380 ","End":"03:44.640","Text":"Let\u0027s see what we get."},{"Start":"03:44.640 ","End":"03:49.605","Text":"5 plus 3/4 is 8/4 is 2,"},{"Start":"03:49.605 ","End":"03:54.780","Text":"and 5 minus 3/4 is 2/4 is 1/2."},{"Start":"03:54.780 ","End":"03:58.635","Text":"First of all, I\u0027m cleared with y not equal to 0."},{"Start":"03:58.635 ","End":"04:02.300","Text":"That\u0027s okay. These are my 2 solutions for y,"},{"Start":"04:02.300 ","End":"04:04.700","Text":"but remember it\u0027s left to get back to x."},{"Start":"04:04.700 ","End":"04:07.395","Text":"According to this, y is 2^x."},{"Start":"04:07.395 ","End":"04:08.970","Text":"I have 2 possibilities."},{"Start":"04:08.970 ","End":"04:12.135","Text":"I have 2^x can either be 2 or 1/2."},{"Start":"04:12.135 ","End":"04:14.850","Text":"Let\u0027s say 2^x equals 2,"},{"Start":"04:14.850 ","End":"04:19.150","Text":"or 2^x equals 1/2."},{"Start":"04:19.150 ","End":"04:21.905","Text":"That I went to write each of these as 2 to the power of something."},{"Start":"04:21.905 ","End":"04:27.605","Text":"2 is just 2^1 and 1/2 is 2^negative 1."},{"Start":"04:27.605 ","End":"04:29.485","Text":"This formula, it\u0027s one over,"},{"Start":"04:29.485 ","End":"04:32.629","Text":"one over something means you make it the power negative."},{"Start":"04:32.629 ","End":"04:34.760","Text":"Now we have everything in terms of base 2,"},{"Start":"04:34.760 ","End":"04:36.020","Text":"so I can compare."},{"Start":"04:36.020 ","End":"04:40.190","Text":"This one gives me that x equals 1."},{"Start":"04:40.190 ","End":"04:44.925","Text":"This one gives me that x equals minus 1."},{"Start":"04:44.925 ","End":"04:47.570","Text":"I have 2 solutions for the original equation,"},{"Start":"04:47.570 ","End":"04:50.615","Text":"x equals 1 or x equals minus 1."},{"Start":"04:50.615 ","End":"04:52.880","Text":"That\u0027s part a."},{"Start":"04:52.880 ","End":"04:56.130","Text":"Let\u0027s move on to part b."},{"Start":"04:56.230 ","End":"05:00.170","Text":"Part b. What do we have here?"},{"Start":"05:00.170 ","End":"05:01.940","Text":"In some ways similar to a,"},{"Start":"05:01.940 ","End":"05:04.055","Text":"except that here we have 4 instead of 2."},{"Start":"05:04.055 ","End":"05:08.055","Text":"What I notice is I have 4^x roughly and other stuff,"},{"Start":"05:08.055 ","End":"05:11.430","Text":"and here I have 4^x and a bit of other stuff."},{"Start":"05:11.430 ","End":"05:16.735","Text":"Let\u0027s see if we can get an equation in just 4^x and 4^ minus x."},{"Start":"05:16.735 ","End":"05:22.420","Text":"We\u0027ll follow in the footsteps of a and get a disguised quadratic and so on."},{"Start":"05:22.420 ","End":"05:26.270","Text":"I\u0027m going to use this formula again"},{"Start":"05:26.270 ","End":"05:32.090","Text":"because here I have a plus 1 and 1/2 and here I have a plus 2."},{"Start":"05:32.090 ","End":"05:38.030","Text":"We\u0027re just saying that I\u0027m viewing this exponent as minus x plus 2."},{"Start":"05:38.030 ","End":"05:39.980","Text":"I can use the plus 1/2 plus 2."},{"Start":"05:39.980 ","End":"05:42.170","Text":"I can use the plus formula."},{"Start":"05:42.170 ","End":"05:51.255","Text":"Here. I have 4^x times 4^1.5 minus 2 times."},{"Start":"05:51.255 ","End":"05:53.385","Text":"Here I have, looking at this,"},{"Start":"05:53.385 ","End":"06:00.905","Text":"4^minus x times 4^2, this equals 60."},{"Start":"06:00.905 ","End":"06:04.230","Text":"Now let us take care of these constants."},{"Start":"06:04.480 ","End":"06:06.980","Text":"4^1.5. I\u0027ll do it at the side."},{"Start":"06:06.980 ","End":"06:09.545","Text":"This is 4^1.5."},{"Start":"06:09.545 ","End":"06:14.270","Text":"We\u0027ll write it as 1.5 or better still, 1 plus 1/2."},{"Start":"06:14.270 ","End":"06:17.195","Text":"Now I can use this formula again."},{"Start":"06:17.195 ","End":"06:21.860","Text":"This is 4^1 times 4^ of 1/2."},{"Start":"06:21.860 ","End":"06:26.585","Text":"For the 1/2, I\u0027m going to use this and convert it to a square root."},{"Start":"06:26.585 ","End":"06:30.005","Text":"This equals 4 times square root of 4,"},{"Start":"06:30.005 ","End":"06:31.280","Text":"where the square root of 4 is 2,"},{"Start":"06:31.280 ","End":"06:33.515","Text":"it\u0027s 4 times 2 is 8."},{"Start":"06:33.515 ","End":"06:36.605","Text":"That takes care of this bit here."},{"Start":"06:36.605 ","End":"06:38.810","Text":"Now here, I also have constants."},{"Start":"06:38.810 ","End":"06:40.535","Text":"I have this and this."},{"Start":"06:40.535 ","End":"06:46.235","Text":"So 2 times4^2 is 2 times 16 is 32,"},{"Start":"06:46.235 ","End":"06:52.265","Text":"okay, now I can rewrite this as 4^1 and 1/2 is 8 and I\u0027ll put the 8 in front."},{"Start":"06:52.265 ","End":"06:56.825","Text":"It\u0027s 8 times 4^x minus,"},{"Start":"06:56.825 ","End":"07:02.245","Text":"this is 32 times 4^ minus x."},{"Start":"07:02.245 ","End":"07:05.535","Text":"This equals 60,"},{"Start":"07:05.535 ","End":"07:08.310","Text":"is there anything we can cancel by yeah,"},{"Start":"07:08.310 ","End":"07:11.570","Text":"4 goes into everything. Let\u0027s rewrite this."},{"Start":"07:11.570 ","End":"07:14.370","Text":"After dividing by 4,"},{"Start":"07:14.370 ","End":"07:20.610","Text":"2 times 4^x minus 32/4 is 8."},{"Start":"07:20.610 ","End":"07:22.550","Text":"And instead of 4^ minus x,"},{"Start":"07:22.550 ","End":"07:24.080","Text":"just like in the previous exercise,"},{"Start":"07:24.080 ","End":"07:28.445","Text":"would write it as 1/4^x using this formula here."},{"Start":"07:28.445 ","End":"07:32.105","Text":"60 divided by 4 is 15."},{"Start":"07:32.105 ","End":"07:33.995","Text":"We\u0027re running out of space,"},{"Start":"07:33.995 ","End":"07:37.430","Text":"let\u0027s scroll up a bit, that\u0027s better."},{"Start":"07:37.430 ","End":"07:39.655","Text":"Now we can continue."},{"Start":"07:39.655 ","End":"07:41.930","Text":"I\u0027m going to use the same thing as we did before."},{"Start":"07:41.930 ","End":"07:47.000","Text":"We notice that this is actually an equation in 4^x."},{"Start":"07:47.000 ","End":"07:51.310","Text":"We can substitute y equals 4^x."},{"Start":"07:51.310 ","End":"07:55.145","Text":"If we do that, then we get an equation in y,"},{"Start":"07:55.145 ","End":"08:03.305","Text":"2y minus, let\u0027s write this as 8/y equals 15."},{"Start":"08:03.305 ","End":"08:11.060","Text":"This is one of those pre quadratic if I multiply both sides by y,"},{"Start":"08:11.060 ","End":"08:15.635","Text":"and later on I have to remember that y cannot be 0."},{"Start":"08:15.635 ","End":"08:19.700","Text":"Make a note to the y naught 0, make a note."},{"Start":"08:19.700 ","End":"08:25.010","Text":"Then we get 2 y times y is 2y^2."},{"Start":"08:25.010 ","End":"08:29.840","Text":"Then let\u0027s just bring this 15y to the other side."},{"Start":"08:29.840 ","End":"08:32.615","Text":"We\u0027ve done this before, minus 15y."},{"Start":"08:32.615 ","End":"08:38.120","Text":"Then here we have minus 8/y times y is just minus 8 equals 0."},{"Start":"08:38.120 ","End":"08:42.485","Text":"Just save the step by bringing the 15y over to the middle."},{"Start":"08:42.485 ","End":"08:46.490","Text":"Quadratic in y, y equals minus b,"},{"Start":"08:46.490 ","End":"08:51.680","Text":"that\u0027s plus 15 plus or minus the square root b^2,"},{"Start":"08:51.680 ","End":"09:01.140","Text":"15 times 15 is 225 minus 4ac minus 4 times 2 times minus 8."},{"Start":"09:01.140 ","End":"09:03.810","Text":"All this over 2a,"},{"Start":"09:03.810 ","End":"09:06.730","Text":"2 times 2, we can already write that as 4."},{"Start":"09:06.730 ","End":"09:16.350","Text":"Let\u0027s see what\u0027s under the square root sign.4 times 2 times 8 is 8 times 8 is 64,"},{"Start":"09:16.350 ","End":"09:21.270","Text":"225 plus 64 is 289."},{"Start":"09:21.270 ","End":"09:24.770","Text":"We have here the square root of 299."},{"Start":"09:24.770 ","End":"09:29.900","Text":"I don\u0027t need a calculator happened to remember the 289 is 17^2."},{"Start":"09:29.900 ","End":"09:32.405","Text":"The square root of 289 is 17."},{"Start":"09:32.405 ","End":"09:33.905","Text":"Do it on your calculator,"},{"Start":"09:33.905 ","End":"09:36.980","Text":"or just multiply 17 times 17 and check."},{"Start":"09:36.980 ","End":"09:45.730","Text":"What we get is 15 plus or minus 17/4."},{"Start":"09:45.730 ","End":"09:54.585","Text":"If we take the plus 15 plus 17 is 32/4 is 8."},{"Start":"09:54.585 ","End":"10:03.240","Text":"If we take the minus 15 minus 17 minus 2 minus 2/4 minus 1/2."},{"Start":"10:03.240 ","End":"10:05.460","Text":"This, as you recall,"},{"Start":"10:05.460 ","End":"10:08.270","Text":"is not x, it is y."},{"Start":"10:08.270 ","End":"10:12.370","Text":"We need to get back to x and we\u0027re going to use this."},{"Start":"10:12.370 ","End":"10:14.440","Text":"We have 2 possibilities,"},{"Start":"10:14.440 ","End":"10:18.790","Text":"either 4^x is 8,"},{"Start":"10:18.790 ","End":"10:22.630","Text":"or 4^x is minus 1/2."},{"Start":"10:22.630 ","End":"10:24.625","Text":"Now we\u0027ve discussed this before."},{"Start":"10:24.625 ","End":"10:29.005","Text":"Positive number to any power is always positive."},{"Start":"10:29.005 ","End":"10:31.420","Text":"This case is ruled out,"},{"Start":"10:31.420 ","End":"10:33.400","Text":"we only have this case."},{"Start":"10:33.400 ","End":"10:35.380","Text":"We only have this possibility."},{"Start":"10:35.380 ","End":"10:37.055","Text":"4^x is 8."},{"Start":"10:37.055 ","End":"10:40.040","Text":"Now it us look like a tough one because 4^1 is 4,"},{"Start":"10:40.040 ","End":"10:42.230","Text":"and 4^2 is 16 we\u0027ve missed."},{"Start":"10:42.230 ","End":"10:43.610","Text":"Somewhere between 1 and 2."},{"Start":"10:43.610 ","End":"10:47.490","Text":"But luckily I remember and I haven\u0027t erased yet."},{"Start":"10:47.490 ","End":"10:53.610","Text":"That 4^1 and 1/2 is 8."},{"Start":"10:53.610 ","End":"10:59.875","Text":"I can write this 8 as 4^1.5 or 1 and 1/2."},{"Start":"10:59.875 ","End":"11:02.375","Text":"It was originally written as 1.5,"},{"Start":"11:02.375 ","End":"11:04.760","Text":"so I\u0027ll go with 1.5."},{"Start":"11:04.760 ","End":"11:07.360","Text":"Now we have equal basis,"},{"Start":"11:07.360 ","End":"11:09.445","Text":"this 4 and this 4."},{"Start":"11:09.445 ","End":"11:16.850","Text":"Then we can compare exponents and get immediately x equals 1.5 or 1 and 1/2."},{"Start":"11:16.850 ","End":"11:19.380","Text":"This is our answer."}],"ID":8135},{"Watched":false,"Name":"Exercise 5","Duration":"15m 26s","ChapterTopicVideoID":8042,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8042.jpeg","UploadDate":"2020-09-30T14:35:48.9230000","DurationForVideoObject":"PT15M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.055","Text":"We have here a pair of exponential equations to solve and we have the formulas if needed."},{"Start":"00:08.055 ","End":"00:11.250","Text":"Let\u0027s begin with a. I examine it,"},{"Start":"00:11.250 ","End":"00:14.130","Text":"and I look and I see 3,"},{"Start":"00:14.130 ","End":"00:15.885","Text":"9, 27,"},{"Start":"00:15.885 ","End":"00:17.655","Text":"all powers of 3."},{"Start":"00:17.655 ","End":"00:22.470","Text":"I see there\u0027s an x in the exponent here, it\u0027s 2x."},{"Start":"00:22.470 ","End":"00:26.325","Text":"I\u0027m thinking 3^x is really where it\u0027s at."},{"Start":"00:26.325 ","End":"00:30.950","Text":"Let\u0027s try and get everything in terms of 3^x,"},{"Start":"00:30.950 ","End":"00:35.775","Text":"27 is 3^3, it\u0027s (3^3)^x."},{"Start":"00:35.775 ","End":"00:40.695","Text":"Then, here we have 3^(2x) plus 1,"},{"Start":"00:40.695 ","End":"00:43.650","Text":"and I can write this as well."},{"Start":"00:43.650 ","End":"00:44.970","Text":"Tell you which formula I\u0027m going to use,"},{"Start":"00:44.970 ","End":"00:48.875","Text":"I\u0027m going to use this one with m being 2x and n being 1."},{"Start":"00:48.875 ","End":"00:50.300","Text":"While we\u0027re at it,"},{"Start":"00:50.300 ","End":"00:52.100","Text":"we obviously going to need this one."},{"Start":"00:52.100 ","End":"00:53.570","Text":"This is our most useful one,"},{"Start":"00:53.570 ","End":"00:57.595","Text":"but we usually use it from right to left, back here."},{"Start":"00:57.595 ","End":"01:06.785","Text":"We\u0027re going rewrite this as 3^(2x) times 3^1 plus becomes a times by this formula,"},{"Start":"01:06.785 ","End":"01:08.900","Text":"and then this equals 9."},{"Start":"01:08.900 ","End":"01:11.725","Text":"This 9 I\u0027ll write as 3^2,"},{"Start":"01:11.725 ","End":"01:16.970","Text":"and this here I\u0027ll also use this formula."},{"Start":"01:16.970 ","End":"01:22.240","Text":"We get 3^x times 3^2."},{"Start":"01:22.240 ","End":"01:24.095","Text":"Let\u0027s see what we can do here."},{"Start":"01:24.095 ","End":"01:27.110","Text":"I said you wanted to put everything in terms of 3^x."},{"Start":"01:27.110 ","End":"01:34.180","Text":"Here, we use a standard trick, (3^3)^x is 3^(3x)."},{"Start":"01:34.180 ","End":"01:38.625","Text":"But if I had written x times 3 or 3 times x, it will be the same."},{"Start":"01:38.625 ","End":"01:44.875","Text":"Actually, you can switch the positions and say that this is (3^x^)3."},{"Start":"01:44.875 ","End":"01:51.470","Text":"You see they both give the same results 3^(3x) or 3^(x3). It\u0027s the same thing."},{"Start":"01:51.470 ","End":"01:55.535","Text":"3^(2x), rather same trick is (3^x)^2."},{"Start":"01:55.535 ","End":"01:58.650","Text":"I could write 2x as x2."},{"Start":"01:58.650 ","End":"02:04.605","Text":"This is this, times 3 is equal to 9."},{"Start":"02:04.605 ","End":"02:12.290","Text":"Once again, the same trick of writing (3^x)^2 minus and here,"},{"Start":"02:12.290 ","End":"02:14.810","Text":"3 times, just change the order,"},{"Start":"02:14.810 ","End":"02:18.545","Text":"times 3^2 times 3^x."},{"Start":"02:18.545 ","End":"02:21.800","Text":"Now, I just want you to notice how we have 3^x everywhere."},{"Start":"02:21.800 ","End":"02:24.140","Text":"We have it here, we have it here,"},{"Start":"02:24.140 ","End":"02:25.700","Text":"we have it here,"},{"Start":"02:25.700 ","End":"02:27.412","Text":"and we have it here."},{"Start":"02:27.412 ","End":"02:33.230","Text":"This really begs for a substitution for 3^x equals something say y."},{"Start":"02:33.230 ","End":"02:38.165","Text":"That\u0027s what I\u0027m going to do, y=3^x."},{"Start":"02:38.165 ","End":"02:41.225","Text":"Let\u0027s see what this equation becomes."},{"Start":"02:41.225 ","End":"02:42.670","Text":"This is y,"},{"Start":"02:42.670 ","End":"02:48.265","Text":"this is y^3 minus y^2,"},{"Start":"02:48.265 ","End":"02:51.640","Text":"better write the 3 in front, 3(y^2)=9."},{"Start":"02:54.290 ","End":"03:03.990","Text":"Also y^2 and minus 3 times 3^2 is 3 times 9 is 27,"},{"Start":"03:03.990 ","End":"03:06.750","Text":"and this is just y."},{"Start":"03:06.750 ","End":"03:10.670","Text":"Now, let me scroll down a bit,"},{"Start":"03:10.670 ","End":"03:17.795","Text":"and let\u0027s bring everything to the left-hand side and put the y\u0027s in order, y^3."},{"Start":"03:17.795 ","End":"03:25.735","Text":"Let\u0027s see, minus 3y^2 minus 9y^2, that\u0027s minus 12y^2."},{"Start":"03:25.735 ","End":"03:30.660","Text":"Then the minus 27y goes over as plus 27y,"},{"Start":"03:30.660 ","End":"03:33.530","Text":"all this now is equal to 0."},{"Start":"03:33.530 ","End":"03:37.010","Text":"I noticed that y appears in all of them,"},{"Start":"03:37.010 ","End":"03:39.590","Text":"so I\u0027ll take it out as a factor."},{"Start":"03:39.590 ","End":"03:41.390","Text":"It\u0027s a good job because otherwise,"},{"Start":"03:41.390 ","End":"03:43.610","Text":"we\u0027d have a cubic equation,"},{"Start":"03:43.610 ","End":"03:45.680","Text":"^3, of which I got the y,"},{"Start":"03:45.680 ","End":"03:47.120","Text":"we just get a quadratic."},{"Start":"03:47.120 ","End":"03:48.590","Text":"It\u0027s now y^2,"},{"Start":"03:48.590 ","End":"03:50.450","Text":"I reduce all the powers by 1,"},{"Start":"03:50.450 ","End":"03:53.105","Text":"minus 12y, and here,"},{"Start":"03:53.105 ","End":"03:54.800","Text":"just plus 27,"},{"Start":"03:54.800 ","End":"03:58.400","Text":"I\u0027ve taken the y outside is equal to 0."},{"Start":"03:58.400 ","End":"04:02.690","Text":"Now, we have a product of two things that is 0."},{"Start":"04:02.690 ","End":"04:07.970","Text":"Either y is 0 or the other possibility"},{"Start":"04:07.970 ","End":"04:13.625","Text":"is that y^2 minus 12y plus 27 equals 0."},{"Start":"04:13.625 ","End":"04:21.605","Text":"Now, this is actually impossible because y is 3^x,"},{"Start":"04:21.605 ","End":"04:25.220","Text":"and 3 to the power of anything is never 0."},{"Start":"04:25.220 ","End":"04:26.750","Text":"It\u0027s always positive."},{"Start":"04:26.750 ","End":"04:28.670","Text":"This is never 0,"},{"Start":"04:28.670 ","End":"04:35.275","Text":"I can immediately rule out this one as a bad possibility for y."},{"Start":"04:35.275 ","End":"04:37.575","Text":"It solves this equation all right,"},{"Start":"04:37.575 ","End":"04:40.140","Text":"but we can\u0027t get back to x from here."},{"Start":"04:40.140 ","End":"04:43.894","Text":"Let\u0027s solve this quadratic equation in y."},{"Start":"04:43.894 ","End":"04:49.715","Text":"We get from the formula y equals minus b is 12"},{"Start":"04:49.715 ","End":"04:55.675","Text":"plus or minus the square root b^2 is 144,"},{"Start":"04:55.675 ","End":"04:58.350","Text":"minus 4 times a,"},{"Start":"04:58.350 ","End":"05:01.650","Text":"which is 1, times c, which is 27."},{"Start":"05:01.650 ","End":"05:04.085","Text":"Then over 2a,"},{"Start":"05:04.085 ","End":"05:06.140","Text":"2 times 1 is 2."},{"Start":"05:06.140 ","End":"05:14.730","Text":"Let\u0027s see now, 4 times 1 times 27 is just 4 times 27 is 108,"},{"Start":"05:16.670 ","End":"05:23.460","Text":"144 minus a 108 gives me 36."},{"Start":"05:23.460 ","End":"05:26.525","Text":"I have here the square root of 36,"},{"Start":"05:26.525 ","End":"05:29.740","Text":"and it comes out to be a whole number, it is 6."},{"Start":"05:29.740 ","End":"05:37.120","Text":"We have here 12 plus or minus 6 over 2,"},{"Start":"05:37.120 ","End":"05:39.555","Text":"and let\u0027s see what possibilities we have,"},{"Start":"05:39.555 ","End":"05:45.990","Text":"12 plus 6 is 18 over 2 is 9,"},{"Start":"05:45.990 ","End":"05:51.300","Text":"and 12 minus 6 is 6 over 2 is 3."},{"Start":"05:51.300 ","End":"05:53.580","Text":"These are the solutions for y,"},{"Start":"05:53.580 ","End":"05:54.885","Text":"not for x,"},{"Start":"05:54.885 ","End":"05:57.015","Text":"and y is 3^x,"},{"Start":"05:57.015 ","End":"05:58.895","Text":"so we have two possibilities."},{"Start":"05:58.895 ","End":"06:04.085","Text":"Either 3^x is 9 and let us see in a moment what that brings us,"},{"Start":"06:04.085 ","End":"06:07.075","Text":"All 3^x equals 3,"},{"Start":"06:07.075 ","End":"06:08.945","Text":"and we\u0027ll see where that goes."},{"Start":"06:08.945 ","End":"06:14.209","Text":"Well, this means that 3^x equals electrolyte 9 is 3 to the something."},{"Start":"06:14.209 ","End":"06:16.190","Text":"Obviously, there\u0027s a 2 here,"},{"Start":"06:16.190 ","End":"06:19.595","Text":"and I wanted to write 3 as 3 to the something,"},{"Start":"06:19.595 ","End":"06:21.440","Text":"I put it as 3^1."},{"Start":"06:21.440 ","End":"06:23.320","Text":"Now I have the same base,"},{"Start":"06:23.320 ","End":"06:25.365","Text":"3 here and here."},{"Start":"06:25.365 ","End":"06:27.645","Text":"This gives me x=2,"},{"Start":"06:27.645 ","End":"06:31.100","Text":"and also I can compare here and here. I\u0027ll write it."},{"Start":"06:31.100 ","End":"06:33.050","Text":"This one is x=2,"},{"Start":"06:33.050 ","End":"06:37.280","Text":"and this gives us x=1, comparing the exponents."},{"Start":"06:37.280 ","End":"06:40.760","Text":"These are the two solutions for x,"},{"Start":"06:40.760 ","End":"06:44.615","Text":"x=1, or x=2."},{"Start":"06:44.615 ","End":"06:45.950","Text":"Now from time to time,"},{"Start":"06:45.950 ","End":"06:48.200","Text":"we like to check our solutions."},{"Start":"06:48.200 ","End":"06:49.580","Text":"Let\u0027s check one of them."},{"Start":"06:49.580 ","End":"06:53.735","Text":"Let\u0027s check that x=1 is a solution and verify it."},{"Start":"06:53.735 ","End":"06:58.705","Text":"I\u0027m going to put x=1 in this equation and verify it."},{"Start":"06:58.705 ","End":"07:00.380","Text":"To verify it, I\u0027ll tell you what I\u0027ll do."},{"Start":"07:00.380 ","End":"07:01.760","Text":"I\u0027ll check the left-hand side."},{"Start":"07:01.760 ","End":"07:04.745","Text":"We sometimes abbreviate left-hand side as LHS,"},{"Start":"07:04.745 ","End":"07:07.460","Text":"until the right of the equals I have the right-hand side."},{"Start":"07:07.460 ","End":"07:10.130","Text":"I\u0027ll substitute in each and see that we get the same thing."},{"Start":"07:10.130 ","End":"07:12.740","Text":"Let\u0027s, first of all, do the left-hand side."},{"Start":"07:12.740 ","End":"07:17.450","Text":"The left-hand side gives us 27^1."},{"Start":"07:17.450 ","End":"07:21.560","Text":"We\u0027re checking this x=1,"},{"Start":"07:21.560 ","End":"07:28.950","Text":"27^1 minus 3 to the twice 1 plus 1 is 3."},{"Start":"07:28.950 ","End":"07:35.807","Text":"27 minus 3^3 is 27 minus,"},{"Start":"07:35.807 ","End":"07:37.420","Text":"3^3 is 27, we get 0,"},{"Start":"07:37.420 ","End":"07:39.295","Text":"that\u0027s the left-hand side."},{"Start":"07:39.295 ","End":"07:41.620","Text":"Now let\u0027s check the right-hand side."},{"Start":"07:41.620 ","End":"07:46.720","Text":"The right-hand side gives us 9 times 9^1,"},{"Start":"07:46.720 ","End":"07:54.249","Text":"x is 1, minus 3 times 3^1 plus 2."},{"Start":"07:54.249 ","End":"07:56.920","Text":"That equals, let\u0027s see,"},{"Start":"07:56.920 ","End":"08:02.748","Text":"3^1 plus 2 is 3^3 is 27,"},{"Start":"08:02.748 ","End":"08:07.780","Text":"we get 9 times 9 is 81 minus 3 times 27,"},{"Start":"08:07.780 ","End":"08:13.600","Text":"that\u0027s also 81 equals 0, these are equal."},{"Start":"08:13.600 ","End":"08:19.450","Text":"Yes, x=1 has being verified as a solution and I\u0027ll let it go with that."},{"Start":"08:19.450 ","End":"08:21.023","Text":"Let\u0027s move on to part b,"},{"Start":"08:21.023 ","End":"08:25.570","Text":"better scroll down a bit and I\u0027ll bring the formulas."},{"Start":"08:25.570 ","End":"08:28.615","Text":"You never know when the formulas might come in handy."},{"Start":"08:28.615 ","End":"08:30.699","Text":"Somehow part a is showing,"},{"Start":"08:30.699 ","End":"08:33.550","Text":"let me just separate that."},{"Start":"08:33.550 ","End":"08:36.520","Text":"That belonged to part a, not relevant."},{"Start":"08:36.520 ","End":"08:38.320","Text":"Now, in part b,"},{"Start":"08:38.320 ","End":"08:42.160","Text":"part b is not something we\u0027ve quite seen before."},{"Start":"08:42.160 ","End":"08:44.310","Text":"We have here 6^x,"},{"Start":"08:44.310 ","End":"08:45.563","Text":"we have 4^x,"},{"Start":"08:45.563 ","End":"08:47.315","Text":"and we have 9^x."},{"Start":"08:47.315 ","End":"08:50.095","Text":"Notice that 4 is 2 times 2,"},{"Start":"08:50.095 ","End":"08:51.970","Text":"9 is 3 times 3,"},{"Start":"08:51.970 ","End":"08:53.905","Text":"and 6 is 2 times 3."},{"Start":"08:53.905 ","End":"08:58.675","Text":"Let\u0027s start off by writing everything in terms of 2\u0027s and 3\u0027s."},{"Start":"08:58.675 ","End":"09:03.760","Text":"What we have here is 5 times."},{"Start":"09:03.760 ","End":"09:05.740","Text":"Now I\u0027m going to use the formula,"},{"Start":"09:05.740 ","End":"09:09.370","Text":"this one, where a is 3 and b is 2."},{"Start":"09:09.370 ","End":"09:12.325","Text":"I can think of 6 as 3 times 2."},{"Start":"09:12.325 ","End":"09:15.370","Text":"If I keep thinking of 6 as 3 times 2 and looking here,"},{"Start":"09:15.370 ","End":"09:19.645","Text":"I\u0027ll get 3^x times 2^x."},{"Start":"09:19.645 ","End":"09:22.135","Text":"Next term, minus 3."},{"Start":"09:22.135 ","End":"09:31.522","Text":"Now 4, I could do the same thing is 2 times 2 and write it as 2^x times 2^x,"},{"Start":"09:31.522 ","End":"09:37.015","Text":"or I could think of it as (2^2)^x and write it as (2^x)^2."},{"Start":"09:37.015 ","End":"09:40.498","Text":"Maybe we will go with this approach."},{"Start":"09:40.498 ","End":"09:42.732","Text":"Here, I have minus 2,"},{"Start":"09:42.732 ","End":"09:46.440","Text":"9^x is like 3^x, 3^x,"},{"Start":"09:46.440 ","End":"09:51.700","Text":"so it\u0027s (3^x)^2 and all this equals 0."},{"Start":"09:51.700 ","End":"09:54.490","Text":"Now, I have two entities here;"},{"Start":"09:54.490 ","End":"09:57.680","Text":"I have 2^x and I have 3^x."},{"Start":"09:57.680 ","End":"10:00.145","Text":"How do we go about this?"},{"Start":"10:00.145 ","End":"10:05.230","Text":"Well, the trick is to divide by one of these,"},{"Start":"10:05.230 ","End":"10:07.750","Text":"either (2^x)^2 or (3^x)^2,"},{"Start":"10:07.750 ","End":"10:09.040","Text":"it doesn\u0027t really matter,"},{"Start":"10:09.040 ","End":"10:13.045","Text":"but I\u0027m going to pick on this one and divide both sides."},{"Start":"10:13.045 ","End":"10:17.875","Text":"I\u0027m dividing both sides by (3^x)^2."},{"Start":"10:17.875 ","End":"10:22.777","Text":"Let\u0027s see, if I divide by (3^x)^2 here."},{"Start":"10:22.777 ","End":"10:28.090","Text":"Well, perhaps I\u0027ll just write it first of all and then afterwards we\u0027ll simplify it."},{"Start":"10:28.090 ","End":"10:29.673","Text":"Here, (3^x)^2,"},{"Start":"10:29.673 ","End":"10:31.060","Text":"and you know what?"},{"Start":"10:31.060 ","End":"10:32.230","Text":"I like it even simpler."},{"Start":"10:32.230 ","End":"10:33.909","Text":"Instead of (3^x)^2,"},{"Start":"10:33.909 ","End":"10:37.495","Text":"I\u0027ll write it as 3^x times 3^x,"},{"Start":"10:37.495 ","End":"10:45.520","Text":"and here I also divide by 3^x times 3^x."},{"Start":"10:45.520 ","End":"10:51.235","Text":"Here also, I\u0027ll divide by (3^x)^2 or 3^x, 3^x."},{"Start":"10:51.235 ","End":"10:53.665","Text":"Let\u0027s see if this makes things a bit easier."},{"Start":"10:53.665 ","End":"10:55.135","Text":"In the first term,"},{"Start":"10:55.135 ","End":"11:00.400","Text":"I can cancel this 3^x with this 3^x,"},{"Start":"11:00.400 ","End":"11:02.965","Text":"in the second term, there\u0027s nothing to cancel,"},{"Start":"11:02.965 ","End":"11:04.719","Text":"and in the third term,"},{"Start":"11:04.719 ","End":"11:07.105","Text":"I can cancel (3^x)^2."},{"Start":"11:07.105 ","End":"11:09.940","Text":"This is 3^x times 3^x."},{"Start":"11:09.940 ","End":"11:14.215","Text":"What I\u0027m left with is 5 times,"},{"Start":"11:14.215 ","End":"11:15.790","Text":"and I\u0027ll write this separately,"},{"Start":"11:15.790 ","End":"11:17.590","Text":"2^x over 3^x,"},{"Start":"11:17.590 ","End":"11:20.074","Text":"since I want to get the 2/3,"},{"Start":"11:20.074 ","End":"11:25.750","Text":"my idea is to approach it and to get 2/3^x."},{"Start":"11:25.750 ","End":"11:29.574","Text":"Here, also I\u0027ll write the 3 out in front."},{"Start":"11:29.574 ","End":"11:33.670","Text":"If I imagine it as 2^x times 2^x."},{"Start":"11:33.670 ","End":"11:38.155","Text":"Instead of this, I\u0027ll just think of it as 2^x times 2^x."},{"Start":"11:38.155 ","End":"11:42.790","Text":"Then what I get is 2^x over 3^x twice,"},{"Start":"11:42.790 ","End":"11:44.560","Text":"I can write it squared or I can just,"},{"Start":"11:44.560 ","End":"11:46.180","Text":"I\u0027ll write it again,"},{"Start":"11:46.180 ","End":"11:49.315","Text":"2^x over 3^x, doesn\u0027t matter."},{"Start":"11:49.315 ","End":"11:53.920","Text":"Here, I have just a number which is 2,"},{"Start":"11:53.920 ","End":"11:56.755","Text":"and then this equals 0."},{"Start":"11:56.755 ","End":"12:00.070","Text":"I think you see where I am headed with this."},{"Start":"12:00.070 ","End":"12:03.865","Text":"If I look at this 2^x over 3^x,"},{"Start":"12:03.865 ","End":"12:05.455","Text":"I have this here,"},{"Start":"12:05.455 ","End":"12:07.300","Text":"and I have it here,"},{"Start":"12:07.300 ","End":"12:11.440","Text":"and here, so I can substitute it."},{"Start":"12:11.440 ","End":"12:17.410","Text":"Say that y=2^x over 3^x."},{"Start":"12:17.410 ","End":"12:21.715","Text":"By the way, notice that if I use this formula,"},{"Start":"12:21.715 ","End":"12:30.285","Text":"it might be better to write 2^x over 3^x as 2/3^x using this,"},{"Start":"12:30.285 ","End":"12:33.600","Text":"whichever one will suit us better later on."},{"Start":"12:33.600 ","End":"12:41.005","Text":"In any event, we get 5 times y minus 3y,"},{"Start":"12:41.005 ","End":"12:45.490","Text":"y, or what we should have written it as y^2 minus 2 equals 0."},{"Start":"12:45.490 ","End":"12:49.480","Text":"Never mind, I\u0027ll rearrange it now and write it as minus"},{"Start":"12:49.480 ","End":"12:57.760","Text":"3y^2 plus 5y minus 2 equals 0."},{"Start":"12:57.760 ","End":"13:02.290","Text":"Now, we have a quadratic equation in y to solve."},{"Start":"13:02.290 ","End":"13:03.790","Text":"But if you\u0027ll indulge me,"},{"Start":"13:03.790 ","End":"13:07.840","Text":"I like to have the coefficient of y^2 to be positive, in other words,"},{"Start":"13:07.840 ","End":"13:12.400","Text":"I like a positive a, so I\u0027m going to multiply both sides by minus 1."},{"Start":"13:12.400 ","End":"13:21.040","Text":"Just make everything minus and I\u0027ve got plus 3y^2 minus 5y plus 2 equals 0."},{"Start":"13:21.040 ","End":"13:24.669","Text":"I found it works better if the a is positive, less mistakes."},{"Start":"13:24.669 ","End":"13:26.814","Text":"Now I\u0027m going to use the formula,"},{"Start":"13:26.814 ","End":"13:30.053","Text":"y equals minus b,"},{"Start":"13:30.053 ","End":"13:32.305","Text":"that\u0027s plus 5,"},{"Start":"13:32.305 ","End":"13:36.370","Text":"plus or minus the square root of b^2,"},{"Start":"13:36.370 ","End":"13:40.780","Text":"25 minus 4ac,"},{"Start":"13:40.780 ","End":"13:43.893","Text":"4 times 3 times 2,"},{"Start":"13:43.893 ","End":"13:47.024","Text":"all these over 2a,"},{"Start":"13:47.024 ","End":"13:48.820","Text":"2 times 3 is 6."},{"Start":"13:48.820 ","End":"13:51.445","Text":"Let\u0027s see what we have under the square root sign,"},{"Start":"13:51.445 ","End":"13:54.880","Text":"4 times 3 times 2 is 24,"},{"Start":"13:54.880 ","End":"13:58.390","Text":"25 minus 24 is 1,"},{"Start":"13:58.390 ","End":"14:02.150","Text":"and the square root of 1 is 1."},{"Start":"14:02.150 ","End":"14:08.425","Text":"We get 5 plus or minus 1 over 6."},{"Start":"14:08.425 ","End":"14:11.440","Text":"Two possibilities, 5 plus 1,"},{"Start":"14:11.440 ","End":"14:13.780","Text":"which is 6 over 6 is 1,"},{"Start":"14:13.780 ","End":"14:17.005","Text":"and 5 minus 1 is 4,"},{"Start":"14:17.005 ","End":"14:20.785","Text":"4 over 6 is 2/3."},{"Start":"14:20.785 ","End":"14:22.660","Text":"Now remember this is not x,"},{"Start":"14:22.660 ","End":"14:23.965","Text":"this is y,"},{"Start":"14:23.965 ","End":"14:27.925","Text":"y is 1 or y is 2/3."},{"Start":"14:27.925 ","End":"14:32.560","Text":"But y is, I\u0027ll use this form 2/3^x."},{"Start":"14:32.560 ","End":"14:38.965","Text":"I have 2/3^x equals 1"},{"Start":"14:38.965 ","End":"14:45.685","Text":"or 2/3^x equals 2/3."},{"Start":"14:45.685 ","End":"14:49.600","Text":"I want to write each of these as 2/3 to the power of something."},{"Start":"14:49.600 ","End":"14:51.880","Text":"Well, this is 2/3^1,"},{"Start":"14:51.880 ","End":"14:54.861","Text":"because anything to the power of 1 is itself."},{"Start":"14:54.861 ","End":"14:59.290","Text":"This is, using this formula here,"},{"Start":"14:59.290 ","End":"15:04.220","Text":"2/3^0, because anything to the power of 0 is 1."},{"Start":"15:04.220 ","End":"15:08.330","Text":"Now, I have the 2/3 in common everywhere,"},{"Start":"15:08.330 ","End":"15:10.055","Text":"and I won\u0027t bother highlighting it."},{"Start":"15:10.055 ","End":"15:16.220","Text":"This gives us immediately by comparing the exponents that x=0."},{"Start":"15:16.220 ","End":"15:22.400","Text":"This one gives us that x=1 and these are our two solutions for x."},{"Start":"15:22.400 ","End":"15:26.840","Text":"That ends part b and we are done."}],"ID":8136},{"Watched":false,"Name":"Exercise 6","Duration":"13m 32s","ChapterTopicVideoID":8043,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8043.jpeg","UploadDate":"2020-09-30T14:40:29.3300000","DurationForVideoObject":"PT13M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.390","Text":"Here we have another pair of exercises in exponential equations."},{"Start":"00:06.390 ","End":"00:10.800","Text":"I can tell you now that both of them will involve substitution."},{"Start":"00:10.800 ","End":"00:15.120","Text":"It looks like in the first one 3^x will be substituted,"},{"Start":"00:15.120 ","End":"00:16.365","Text":"and in the second one,"},{"Start":"00:16.365 ","End":"00:18.840","Text":"I\u0027m guessing that 4^x is going to be at."},{"Start":"00:18.840 ","End":"00:20.265","Text":"But let\u0027s start with the first."},{"Start":"00:20.265 ","End":"00:24.525","Text":"As usual, I provide different formulas for exponents if we need them."},{"Start":"00:24.525 ","End":"00:29.565","Text":"Let\u0027s try and see if we can get everything in terms of 3^x."},{"Start":"00:29.565 ","End":"00:31.665","Text":"This already is 3^x,"},{"Start":"00:31.665 ","End":"00:35.865","Text":"I want to convert the 9^x in terms of 3^x."},{"Start":"00:35.865 ","End":"00:41.595","Text":"The first part stays as is 1 over 3^x plus 1 minus 2 over."},{"Start":"00:41.595 ","End":"00:46.420","Text":"Now, this is 3^x squared minus 1."},{"Start":"00:46.420 ","End":"00:47.690","Text":"It\u0027s a trick we\u0027ve used often,"},{"Start":"00:47.690 ","End":"00:51.020","Text":"but I\u0027ll remind you at the side here of how we do that."},{"Start":"00:51.020 ","End":"00:54.980","Text":"We write that 9^x is"},{"Start":"00:54.980 ","End":"01:03.425","Text":"3^2 to the power of x. I can change the order of the 2 and the x."},{"Start":"01:03.425 ","End":"01:04.730","Text":"To see even this,"},{"Start":"01:04.730 ","End":"01:08.720","Text":"I\u0027ll show you that this is 3^2x from the formula."},{"Start":"01:08.720 ","End":"01:10.835","Text":"This one that we use most often,"},{"Start":"01:10.835 ","End":"01:14.525","Text":"3^2x is 3^2 to the power of x."},{"Start":"01:14.525 ","End":"01:19.660","Text":"But, I can look at 2x as x times 2 and then"},{"Start":"01:19.660 ","End":"01:25.120","Text":"I can reverse the steps by saying this is 3^x to the power of 2."},{"Start":"01:25.120 ","End":"01:28.580","Text":"But in practice, you would just do this straight away."},{"Start":"01:28.580 ","End":"01:30.980","Text":"You would say 9 is 3^2,"},{"Start":"01:30.980 ","End":"01:35.815","Text":"so 9^x is 3^x all squared and this equals 0."},{"Start":"01:35.815 ","End":"01:42.890","Text":"Now, I have everything in terms of 3^x and I\u0027ll make the substitution y equals 3^x."},{"Start":"01:42.890 ","End":"01:44.435","Text":"Let\u0027s see what we get,"},{"Start":"01:44.435 ","End":"01:46.100","Text":"need a bit more space,"},{"Start":"01:46.100 ","End":"01:49.350","Text":"we\u0027re going to take myself a lot more space."},{"Start":"01:49.760 ","End":"01:51.920","Text":"What we have here,"},{"Start":"01:51.920 ","End":"01:57.560","Text":"after the substitution is 1 over 3^x is y plus 1"},{"Start":"01:57.560 ","End":"02:05.145","Text":"minus 2 over y^2 minus 1 equals 0."},{"Start":"02:05.145 ","End":"02:08.660","Text":"Now, I\u0027d like to make a common denominator for this."},{"Start":"02:08.660 ","End":"02:10.445","Text":"This is a fractions equation."},{"Start":"02:10.445 ","End":"02:11.870","Text":"But before I do that,"},{"Start":"02:11.870 ","End":"02:18.410","Text":"I\u0027d like to point out that this is a difference of squares because y^2 minus 1 really,"},{"Start":"02:18.410 ","End":"02:22.160","Text":"but it\u0027s also y^2 minus 1^2,"},{"Start":"02:22.160 ","End":"02:27.320","Text":"and difference of squares formula tells us that this is y plus"},{"Start":"02:27.320 ","End":"02:33.274","Text":"1 times y minus 1 or the other way round, whichever."},{"Start":"02:33.274 ","End":"02:36.424","Text":"I wasn\u0027t going to do this, but I\u0027ll remind you of the formula,"},{"Start":"02:36.424 ","End":"02:40.480","Text":"a^2 minus b^2 is a plus b,"},{"Start":"02:40.480 ","End":"02:41.810","Text":"a minus b,"},{"Start":"02:41.810 ","End":"02:44.045","Text":"for those who really have forgotten everything."},{"Start":"02:44.045 ","End":"02:47.840","Text":"Now, what I have here is y plus 1,"},{"Start":"02:47.840 ","End":"02:50.750","Text":"and here I have y plus 1y minus 1."},{"Start":"02:50.750 ","End":"02:53.300","Text":"The common denominator can be just this."},{"Start":"02:53.300 ","End":"02:56.255","Text":"This will serve as the denominator for this also."},{"Start":"02:56.255 ","End":"03:02.130","Text":"A usual technique is I\u0027m going to multiply both sides by y plus 1,"},{"Start":"03:02.130 ","End":"03:06.470","Text":"y minus 1 and I\u0027ll write what\u0027s missing in each one here,"},{"Start":"03:06.470 ","End":"03:09.680","Text":"y plus 1 is missing a y minus 1."},{"Start":"03:09.680 ","End":"03:11.150","Text":"Here, I\u0027m not missing anything,"},{"Start":"03:11.150 ","End":"03:14.000","Text":"I multiply by 1 and this side is 0,"},{"Start":"03:14.000 ","End":"03:17.990","Text":"whatever I multiply, it doesn\u0027t matter, it stays 0."},{"Start":"03:17.990 ","End":"03:19.835","Text":"What I get from this one,"},{"Start":"03:19.835 ","End":"03:23.650","Text":"y minus 1 times 1 is just y minus 1,"},{"Start":"03:23.650 ","End":"03:29.360","Text":"minus and here 1 times 2 is 2, and this equals 0."},{"Start":"03:29.360 ","End":"03:31.635","Text":"How convenient it\u0027s very simple now,"},{"Start":"03:31.635 ","End":"03:35.690","Text":"just bring the numbers to the other side and we get on the right-hand side,"},{"Start":"03:35.690 ","End":"03:36.970","Text":"1 plus 2,"},{"Start":"03:36.970 ","End":"03:38.650","Text":"which I know is 3,"},{"Start":"03:38.650 ","End":"03:40.715","Text":"even without a calculator."},{"Start":"03:40.715 ","End":"03:43.880","Text":"This is our answer. Not quite."},{"Start":"03:43.880 ","End":"03:46.070","Text":"It is my answer for y,"},{"Start":"03:46.070 ","End":"03:52.070","Text":"but I\u0027m looking for x. I\u0027ll just go back here and say that y,"},{"Start":"03:52.070 ","End":"03:53.900","Text":"which is 3^x,"},{"Start":"03:53.900 ","End":"03:58.950","Text":"is equal to 3 and 3 is 3^1."},{"Start":"03:58.950 ","End":"04:04.130","Text":"The reason I\u0027m doing that is that now I have two things with the same base 3,"},{"Start":"04:04.130 ","End":"04:08.770","Text":"so I can compare the exponents and get that x equals 1."},{"Start":"04:08.770 ","End":"04:10.950","Text":"This is what I\u0027m looking for."},{"Start":"04:10.950 ","End":"04:14.465","Text":"Let\u0027s quickly check that this indeed works."},{"Start":"04:14.465 ","End":"04:16.849","Text":"This is not a mandatory part of the exercise,"},{"Start":"04:16.849 ","End":"04:19.610","Text":"but from time to time I\u0027ll do it just to make"},{"Start":"04:19.610 ","End":"04:23.235","Text":"sure and I recommend doing it whenever you have the time for it."},{"Start":"04:23.235 ","End":"04:27.530","Text":"Let\u0027s see, putting x equals 1 here, what do I get?"},{"Start":"04:27.530 ","End":"04:34.115","Text":"I get 1 over 3^1 plus 1,"},{"Start":"04:34.115 ","End":"04:40.300","Text":"minus 2 over 9^1 minus 1."},{"Start":"04:40.300 ","End":"04:41.940","Text":"Now from the equation,"},{"Start":"04:41.940 ","End":"04:43.838","Text":"I want to see that I get 0,"},{"Start":"04:43.838 ","End":"04:49.010","Text":"let me expand it and simplify it and hope that at the end I get 0."},{"Start":"04:49.010 ","End":"04:53.570","Text":"Let\u0027s see, 3^1 is 3 plus 1 is 4,"},{"Start":"04:53.570 ","End":"04:55.705","Text":"this is a 1/4."},{"Start":"04:55.705 ","End":"05:02.340","Text":"Here I get 9^1 is 9 minus 1 is 8,"},{"Start":"05:02.340 ","End":"05:10.600","Text":"I have minus 2/8 and a 1/4 minus 2/8."},{"Start":"05:10.600 ","End":"05:13.160","Text":"Certainly, the 2/8 is a 1/4,"},{"Start":"05:13.160 ","End":"05:14.975","Text":"I can divide top and bottom by 2."},{"Start":"05:14.975 ","End":"05:19.480","Text":"Indeed, this is equal to 0."},{"Start":"05:19.480 ","End":"05:24.950","Text":"That means that we have verified this and we\u0027re okay."},{"Start":"05:24.950 ","End":"05:26.360","Text":"Before I go to part b,"},{"Start":"05:26.360 ","End":"05:28.370","Text":"why don\u0027t I just take the formulas with me,"},{"Start":"05:28.370 ","End":"05:30.635","Text":"you never know when I might need them."},{"Start":"05:30.635 ","End":"05:37.595","Text":"Now, here we are in part b. I can see that here it\u0027s all about 4^x."},{"Start":"05:37.595 ","End":"05:41.420","Text":"Again, we want to try a substitution and we want to substitute 4^x."},{"Start":"05:41.420 ","End":"05:46.925","Text":"Let\u0027s just rewrite this using the formulas to make the 4^x stand out more clearly."},{"Start":"05:46.925 ","End":"05:51.680","Text":"Because I have here an exponent x plus 1 and an x minus 1,"},{"Start":"05:51.680 ","End":"05:58.970","Text":"I\u0027ll be wanting to use this formula for the x plus 1 and this formula for the x minus 1."},{"Start":"05:58.970 ","End":"06:05.840","Text":"What we get is 124 over 4^x times"},{"Start":"06:05.840 ","End":"06:15.705","Text":"4^1 minus 4 plus 8 over 4^x divided by,"},{"Start":"06:15.705 ","End":"06:17.685","Text":"I\u0027ll write divided by this way,"},{"Start":"06:17.685 ","End":"06:23.200","Text":"still won\u0027t have a denominator in the denominator and this equals 2."},{"Start":"06:23.200 ","End":"06:27.320","Text":"Now, I want to simplify this a bit but on second thoughts,"},{"Start":"06:27.320 ","End":"06:30.095","Text":"we can do the substitution first and then simplify."},{"Start":"06:30.095 ","End":"06:33.230","Text":"Let\u0027s put in y equals 4^x,"},{"Start":"06:33.230 ","End":"06:34.850","Text":"because we\u0027ve got that here and here."},{"Start":"06:34.850 ","End":"06:38.915","Text":"We\u0027re rewriting this, I\u0027ve got 124 over,"},{"Start":"06:38.915 ","End":"06:40.530","Text":"this is y and this is 4,"},{"Start":"06:40.530 ","End":"06:48.255","Text":"this is 4y minus 4 plus 8 over y divided by 4."},{"Start":"06:48.255 ","End":"06:52.332","Text":"I\u0027ll write the divided by this way, equals 2."},{"Start":"06:52.332 ","End":"06:58.045","Text":"Now, what I\u0027m going to do is here I\u0027m going to put the 4,"},{"Start":"06:58.045 ","End":"06:59.390","Text":"just rewrite this term."},{"Start":"06:59.390 ","End":"07:01.520","Text":"If you divide by a fraction,"},{"Start":"07:01.520 ","End":"07:03.169","Text":"you multiply by the reciprocal."},{"Start":"07:03.169 ","End":"07:07.070","Text":"I just want to rewrite this by putting the 4,"},{"Start":"07:07.070 ","End":"07:09.223","Text":"erasing it from here,"},{"Start":"07:09.223 ","End":"07:13.040","Text":"and putting it into the numerator times 4."},{"Start":"07:13.040 ","End":"07:15.241","Text":"If you\u0027re not sure about this with fractions,"},{"Start":"07:15.241 ","End":"07:20.390","Text":"I can explain it that a divided by b"},{"Start":"07:20.390 ","End":"07:25.790","Text":"divided by c is a times c over b,"},{"Start":"07:25.790 ","End":"07:27.439","Text":"multiplying by the reciprocal,"},{"Start":"07:27.439 ","End":"07:30.305","Text":"and this is just ac over b."},{"Start":"07:30.305 ","End":"07:31.985","Text":"It should have been clear."},{"Start":"07:31.985 ","End":"07:36.141","Text":"Now, I want to take a common denominator for both sides"},{"Start":"07:36.141 ","End":"07:40.540","Text":"but I see now that 124 is divisible by 4."},{"Start":"07:40.540 ","End":"07:43.390","Text":"I think we can simplify a bit more before we do"},{"Start":"07:43.390 ","End":"07:46.855","Text":"a common denominator because the denominator,"},{"Start":"07:46.855 ","End":"07:48.340","Text":"let me do it like this."},{"Start":"07:48.340 ","End":"07:54.185","Text":"This is a 124 and this is 4 times y minus 1."},{"Start":"07:54.185 ","End":"07:55.735","Text":"I\u0027ll write the rest of it out,"},{"Start":"07:55.735 ","End":"07:59.755","Text":"32 over y equals 2."},{"Start":"07:59.755 ","End":"08:01.900","Text":"Here, we can cancel by 4,"},{"Start":"08:01.900 ","End":"08:07.460","Text":"this goes into this 31 times."},{"Start":"08:07.460 ","End":"08:09.930","Text":"Now it\u0027s a good time for the common denominator."},{"Start":"08:09.930 ","End":"08:12.220","Text":"I see I have y minus 1 here and I have y,"},{"Start":"08:12.220 ","End":"08:20.355","Text":"let me multiply both sides by y times y minus 1, the product."},{"Start":"08:20.355 ","End":"08:25.820","Text":"What I\u0027ll need to do here is to multiply by y,"},{"Start":"08:25.820 ","End":"08:26.960","Text":"let me use a different color,"},{"Start":"08:26.960 ","End":"08:31.283","Text":"too much orange there and here by y minus 1."},{"Start":"08:31.283 ","End":"08:33.845","Text":"Here, since this is 2 over 1,"},{"Start":"08:33.845 ","End":"08:38.950","Text":"let me to multiply it by the full y, y minus 1."},{"Start":"08:38.950 ","End":"08:44.205","Text":"We get y times 31 is 31y."},{"Start":"08:44.205 ","End":"08:51.240","Text":"Here, 32 times y minus 1 better than y minus 1 times 32."},{"Start":"08:51.240 ","End":"08:52.575","Text":"Put the number first."},{"Start":"08:52.575 ","End":"08:59.660","Text":"Here also put the number first and then the y, y minus 1."},{"Start":"08:59.660 ","End":"09:04.070","Text":"Next step is to expand the brackets,"},{"Start":"09:04.070 ","End":"09:09.030","Text":"31y plus 32y minus"},{"Start":"09:09.030 ","End":"09:15.926","Text":"32 equals 2y times y is 2y^2 minus 2y."},{"Start":"09:15.926 ","End":"09:19.040","Text":"Quadratic equation, but let\u0027s arrange it."},{"Start":"09:19.040 ","End":"09:20.360","Text":"It\u0027s disarranged at the moment,"},{"Start":"09:20.360 ","End":"09:22.175","Text":"put everything on the left."},{"Start":"09:22.175 ","End":"09:27.605","Text":"Let\u0027s see, I\u0027ll collect y^2 terms first, minus 2y^2."},{"Start":"09:27.605 ","End":"09:29.840","Text":"Let\u0027s see, where do we have y terms?"},{"Start":"09:29.840 ","End":"09:33.440","Text":"We have 31 here plus 32,"},{"Start":"09:33.440 ","End":"09:36.185","Text":"that makes it 63,"},{"Start":"09:36.185 ","End":"09:40.340","Text":"plus another 2 is 65y,"},{"Start":"09:40.340 ","End":"09:47.090","Text":"and numbers just the minus 32 that was here equals 0."},{"Start":"09:47.090 ","End":"09:49.190","Text":"You probably know me already,"},{"Start":"09:49.190 ","End":"09:53.960","Text":"I like to have a positive a so I\u0027m going to multiply everything by minus,"},{"Start":"09:53.960 ","End":"09:56.750","Text":"it\u0027s minus 1 just to reverse all the signs."},{"Start":"09:56.750 ","End":"10:04.385","Text":"2y^2 minus 65y plus 32 equals 0."},{"Start":"10:04.385 ","End":"10:07.715","Text":"Somehow I make less mistakes when it\u0027s a positive a,"},{"Start":"10:07.715 ","End":"10:09.260","Text":"and I recommend it."},{"Start":"10:09.260 ","End":"10:13.760","Text":"Formula ay^2 minus by plus c equals 0,"},{"Start":"10:13.760 ","End":"10:19.055","Text":"y is equal to minus b is"},{"Start":"10:19.055 ","End":"10:25.435","Text":"65 plus or minus the square root of b^2,"},{"Start":"10:25.435 ","End":"10:35.010","Text":"65^2 minus 4 times a is 2 and c is 32,"},{"Start":"10:35.010 ","End":"10:37.110","Text":"this is all over 2a,"},{"Start":"10:37.110 ","End":"10:40.380","Text":"which is 2 times 2."},{"Start":"10:40.380 ","End":"10:41.730","Text":"Let\u0027s see now,"},{"Start":"10:41.730 ","End":"10:46.338","Text":"I think we might need a calculator or at least we\u0027ll do some by calculator."},{"Start":"10:46.338 ","End":"10:49.910","Text":"65 times 65 is"},{"Start":"10:49.910 ","End":"10:58.890","Text":"4,225 and 4 times 2 times 32, this is 256."},{"Start":"10:58.890 ","End":"11:02.325","Text":"I\u0027m going to subtract 256."},{"Start":"11:02.325 ","End":"11:08.865","Text":"This gives me 3,969."},{"Start":"11:08.865 ","End":"11:11.630","Text":"I\u0027m going to also need the square root of this one,"},{"Start":"11:11.630 ","End":"11:12.980","Text":"I don\u0027t know by heart,"},{"Start":"11:12.980 ","End":"11:15.935","Text":"if I take the square root, I get 63."},{"Start":"11:15.935 ","End":"11:25.195","Text":"What we have is 65 plus or minus 63 over 4."},{"Start":"11:25.195 ","End":"11:28.575","Text":"Now, we branch for the plus and for the minus."},{"Start":"11:28.575 ","End":"11:30.651","Text":"For the plus,"},{"Start":"11:30.651 ","End":"11:33.150","Text":"65 plus 63 is 128,"},{"Start":"11:33.150 ","End":"11:37.700","Text":"128 over 4 is 32."},{"Start":"11:37.700 ","End":"11:39.530","Text":"If I take the minus,"},{"Start":"11:39.530 ","End":"11:42.220","Text":"65 minus 63 is 2,"},{"Start":"11:42.220 ","End":"11:45.245","Text":"2 over 4 is 1/2."},{"Start":"11:45.245 ","End":"11:47.960","Text":"Now, these two answers are not the answers for x,"},{"Start":"11:47.960 ","End":"11:49.910","Text":"they are the answers for y."},{"Start":"11:49.910 ","End":"11:53.900","Text":"But y, here it is, is 4^x."},{"Start":"11:53.900 ","End":"12:00.495","Text":"Back here, we have either 4^x is 32,"},{"Start":"12:00.495 ","End":"12:06.705","Text":"or 4^x is equal to 1/2."},{"Start":"12:06.705 ","End":"12:08.714","Text":"Here\u0027s the thing,"},{"Start":"12:08.714 ","End":"12:12.245","Text":"this is not a whole number power of x,"},{"Start":"12:12.245 ","End":"12:17.075","Text":"because 4^2 is 16 and 4^3 is 64,"},{"Start":"12:17.075 ","End":"12:19.055","Text":"and this isn\u0027t a whole number either."},{"Start":"12:19.055 ","End":"12:24.155","Text":"What we can do is just think of everything in terms of powers of 2."},{"Start":"12:24.155 ","End":"12:27.923","Text":"Now, 4^x is 2^2 to the x,"},{"Start":"12:27.923 ","End":"12:31.540","Text":"that\u0027s easily seen to be 2^2x."},{"Start":"12:31.540 ","End":"12:33.920","Text":"The same here, I\u0027ll work on them both."},{"Start":"12:33.920 ","End":"12:35.960","Text":"If you\u0027re not sure about that, I\u0027ll show you again."},{"Start":"12:35.960 ","End":"12:37.798","Text":"It\u0027s the trick we\u0027ve done before,"},{"Start":"12:37.798 ","End":"12:44.084","Text":"4^x is 2^2 to the power of x."},{"Start":"12:44.084 ","End":"12:45.589","Text":"Then we use this formula,"},{"Start":"12:45.589 ","End":"12:48.565","Text":"and this is 2^2x,"},{"Start":"12:48.565 ","End":"12:51.740","Text":"32 is easy to write as a power of 2."},{"Start":"12:51.740 ","End":"12:52.970","Text":"I know this is 2^5,"},{"Start":"12:52.970 ","End":"12:54.304","Text":"we\u0027ve seen this before."},{"Start":"12:54.304 ","End":"12:57.980","Text":"1/2 we\u0027ve also seen before is 2 to the minus 1."},{"Start":"12:57.980 ","End":"13:01.115","Text":"What I get for this one,"},{"Start":"13:01.115 ","End":"13:02.635","Text":"now we have equal bases;"},{"Start":"13:02.635 ","End":"13:07.890","Text":"2 and 2, so we can say 2x is equal to 5."},{"Start":"13:07.890 ","End":"13:13.260","Text":"Finally, from here I get x equals 5 over 2,"},{"Start":"13:13.260 ","End":"13:15.075","Text":"which is 2 and 1/2."},{"Start":"13:15.075 ","End":"13:19.890","Text":"If I take this path 2x is minus 1,"},{"Start":"13:19.890 ","End":"13:24.165","Text":"and so x is minus 1/2."},{"Start":"13:24.165 ","End":"13:33.670","Text":"The solutions for part b is x is either 2 and 1/2 or x is minus 1/2. We\u0027re done."}],"ID":8137},{"Watched":false,"Name":"Exercise 7","Duration":"12m 24s","ChapterTopicVideoID":8044,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8044.jpeg","UploadDate":"2020-09-30T13:54:46.0530000","DurationForVideoObject":"PT12M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"Here we have two equations to solve."},{"Start":"00:02.460 ","End":"00:05.760","Text":"They\u0027re both exponential equations and what\u0027s common"},{"Start":"00:05.760 ","End":"00:09.375","Text":"to them both is that they\u0027ll both use the technique of substitution."},{"Start":"00:09.375 ","End":"00:14.475","Text":"In the first case, we\u0027re going to substitute 5^x and the second 9^x."},{"Start":"00:14.475 ","End":"00:16.230","Text":"Let\u0027s start with the first one."},{"Start":"00:16.230 ","End":"00:21.520","Text":"As usual, I have a table of formulas here in case we need."},{"Start":"00:21.650 ","End":"00:29.700","Text":"I see here that I have the 5^x and here it\u0027s not quite 5^x is 5^x plus 1."},{"Start":"00:29.700 ","End":"00:34.065","Text":"We\u0027ll use our usual trick of writing the x plus 1."},{"Start":"00:34.065 ","End":"00:38.475","Text":"Well, using this formula with m plus n equals x plus 1."},{"Start":"00:38.475 ","End":"00:41.240","Text":"What we get is mostly the same."},{"Start":"00:41.240 ","End":"00:43.265","Text":"The first term is the same,"},{"Start":"00:43.265 ","End":"00:44.750","Text":"but the second term,"},{"Start":"00:44.750 ","End":"00:46.775","Text":"I\u0027m going to replace it by,"},{"Start":"00:46.775 ","End":"00:48.065","Text":"according to this formula,"},{"Start":"00:48.065 ","End":"00:50.994","Text":"5^x times 5^1,"},{"Start":"00:50.994 ","End":"00:53.930","Text":"5^1 is just 5."},{"Start":"00:53.930 ","End":"00:59.270","Text":"I don\u0027t need to write to the power of one and this is equal to 3."},{"Start":"00:59.270 ","End":"01:01.430","Text":"Now we\u0027ll make the substitution,"},{"Start":"01:01.430 ","End":"01:03.230","Text":"5^x appears in both,"},{"Start":"01:03.230 ","End":"01:05.105","Text":"and that\u0027s the only way x appears."},{"Start":"01:05.105 ","End":"01:07.825","Text":"We\u0027ll take another letter y,"},{"Start":"01:07.825 ","End":"01:10.415","Text":"that\u0027s my favorite letter after x,"},{"Start":"01:10.415 ","End":"01:12.880","Text":"is equal to 5^x."},{"Start":"01:12.880 ","End":"01:14.780","Text":"If we substitute it here,"},{"Start":"01:14.780 ","End":"01:15.870","Text":"we will get, just let me get some more space here, there we go."},{"Start":"01:19.000 ","End":"01:25.895","Text":"Then we will get 9/2 minus y,"},{"Start":"01:25.895 ","End":"01:28.370","Text":"minus 2 over,"},{"Start":"01:28.370 ","End":"01:33.735","Text":"I\u0027ll put the five first, 5 times y=3."},{"Start":"01:33.735 ","End":"01:37.436","Text":"Equation with fractions common denominator,"},{"Start":"01:37.436 ","End":"01:41.120","Text":"it\u0027s going to be 5 times y times 2 minus y."},{"Start":"01:41.120 ","End":"01:45.860","Text":"We need everything 5y, 2 minus y."},{"Start":"01:45.860 ","End":"01:48.725","Text":"As usual, we see what\u0027s missing."},{"Start":"01:48.725 ","End":"01:51.110","Text":"2 minus y is missing 5y."},{"Start":"01:51.110 ","End":"01:56.000","Text":"5y is missing 2 minus y and 3,"},{"Start":"01:56.000 ","End":"01:58.745","Text":"which is 3/1 is multiplied by the whole thing."},{"Start":"01:58.745 ","End":"02:02.105","Text":"5y, 2 minus y,"},{"Start":"02:02.105 ","End":"02:06.665","Text":"we get 5y times 9 is 45y."},{"Start":"02:06.665 ","End":"02:11.150","Text":"Here, we have twice 2 minus y,"},{"Start":"02:11.150 ","End":"02:13.925","Text":"prefer to write the 2 first and on the right,"},{"Start":"02:13.925 ","End":"02:17.285","Text":"I\u0027ll combine the 3 with the 5 is 15,"},{"Start":"02:17.285 ","End":"02:21.680","Text":"and we still have y 2 minus y."},{"Start":"02:21.680 ","End":"02:25.910","Text":"A bit of practice in expansion,"},{"Start":"02:25.910 ","End":"02:33.410","Text":"45y as is minus 2 times 2 is 4 and 2 times y,"},{"Start":"02:33.410 ","End":"02:34.580","Text":"but it\u0027s minus minus,"},{"Start":"02:34.580 ","End":"02:37.715","Text":"so it\u0027s plus 2y,"},{"Start":"02:37.715 ","End":"02:42.740","Text":"and this is equal to 15y times 2 is 30y."},{"Start":"02:42.740 ","End":"02:46.685","Text":"Then we have minus 15y^2."},{"Start":"02:46.685 ","End":"02:50.750","Text":"It looks very much like a disarray changed quadratic let us just arrange it,"},{"Start":"02:50.750 ","End":"02:53.030","Text":"put everything on the left."},{"Start":"02:53.030 ","End":"02:54.560","Text":"But let\u0027s do it in order."},{"Start":"02:54.560 ","End":"02:55.932","Text":"Let\u0027s take the y^2 first,"},{"Start":"02:55.932 ","End":"02:57.710","Text":"that\u0027s the only place I have it."},{"Start":"02:57.710 ","End":"02:59.660","Text":"It\u0027s 15y^2."},{"Start":"02:59.660 ","End":"03:02.225","Text":"It\u0027s a plus because it went over to the other side."},{"Start":"03:02.225 ","End":"03:03.860","Text":"Let\u0027s see where do we have y?"},{"Start":"03:03.860 ","End":"03:08.360","Text":"We have 45 and 2 is 47,"},{"Start":"03:08.360 ","End":"03:13.250","Text":"but we have to subtract 30 because this one comes over as minus 47 minus"},{"Start":"03:13.250 ","End":"03:18.680","Text":"30 is plus 17y and numbers,"},{"Start":"03:18.680 ","End":"03:21.343","Text":"that\u0027s just the minus 4 here."},{"Start":"03:21.343 ","End":"03:24.530","Text":"On the right we have nothing left at 0."},{"Start":"03:24.530 ","End":"03:27.710","Text":"Quadratic equation in y formula,"},{"Start":"03:27.710 ","End":"03:33.770","Text":"minus 17 plus or minus the square root of, let\u0027s see,"},{"Start":"03:33.770 ","End":"03:38.570","Text":"17^2 minus 4 times"},{"Start":"03:38.570 ","End":"03:46.430","Text":"15 times minus 4/2a or that part is 2 times 15."},{"Start":"03:46.430 ","End":"03:49.640","Text":"I\u0027ll write it already as 30 twice 15."},{"Start":"03:49.640 ","End":"03:53.225","Text":"What interested me most is what\u0027s under the square root sign."},{"Start":"03:53.225 ","End":"03:59.105","Text":"Let\u0027s see, 17^2 actually memorize is 289."},{"Start":"03:59.105 ","End":"04:00.675","Text":"Or you can use a calculator."},{"Start":"04:00.675 ","End":"04:06.680","Text":"Here, I see a minus and a minus so it\u0027s going to be plus 4 times 15 is 60,"},{"Start":"04:06.680 ","End":"04:10.150","Text":"60 times 4 is 240."},{"Start":"04:10.150 ","End":"04:12.470","Text":"What does that come out to be?"},{"Start":"04:12.470 ","End":"04:20.900","Text":"That comes out as 529 and while I have my calculator out,"},{"Start":"04:20.900 ","End":"04:26.105","Text":"let me take the square root of that it gives me 23."},{"Start":"04:26.105 ","End":"04:28.250","Text":"Let\u0027s go back here."},{"Start":"04:28.250 ","End":"04:36.520","Text":"We can write this as minus 17 plus or minus 23/30."},{"Start":"04:36.520 ","End":"04:39.235","Text":"We branch for the plus and for the minus."},{"Start":"04:39.235 ","End":"04:43.820","Text":"In the case of the plus 23 minus 17 is 6,"},{"Start":"04:43.820 ","End":"04:47.435","Text":"6/30, that\u0027s easy, it\u0027s 1/5."},{"Start":"04:47.435 ","End":"04:49.490","Text":"Let\u0027s do it the other way, minus 17,"},{"Start":"04:49.490 ","End":"04:56.203","Text":"minus 23 is minus 40/30 is minus 4/3,"},{"Start":"04:56.203 ","End":"04:58.505","Text":"or minus 1/3,"},{"Start":"04:58.505 ","End":"05:01.040","Text":"but it\u0027s a negative and you may know from"},{"Start":"05:01.040 ","End":"05:04.175","Text":"experience already that the negative is not going to matter."},{"Start":"05:04.175 ","End":"05:07.580","Text":"Why is that? These other solutions,"},{"Start":"05:07.580 ","End":"05:09.485","Text":"not for x but for y."},{"Start":"05:09.485 ","End":"05:13.160","Text":"These are the solutions for y and y is five to the x,"},{"Start":"05:13.160 ","End":"05:18.665","Text":"then we have a choice that either 5^x is 1/5,"},{"Start":"05:18.665 ","End":"05:23.015","Text":"or 5^x equals minus 4/3."},{"Start":"05:23.015 ","End":"05:25.115","Text":"Now, I\u0027ve mentioned this previously."},{"Start":"05:25.115 ","End":"05:27.215","Text":"Five to the power of anything,"},{"Start":"05:27.215 ","End":"05:30.275","Text":"positive base to the power of anything is always positive."},{"Start":"05:30.275 ","End":"05:32.825","Text":"Never negative, doesn\u0027t matter what x you put here."},{"Start":"05:32.825 ","End":"05:35.855","Text":"This is impossible."},{"Start":"05:35.855 ","End":"05:38.585","Text":"We only have to concentrate on this case."},{"Start":"05:38.585 ","End":"05:43.240","Text":"Now, as usual, one over something is that something to the minus 1,"},{"Start":"05:43.240 ","End":"05:47.090","Text":"and now a 5^x = 5 to the minus 1."},{"Start":"05:47.090 ","End":"05:51.795","Text":"Same base compare exponents and we get that x"},{"Start":"05:51.795 ","End":"05:57.660","Text":"= minus one and that\u0027s the answer to Part A on to Part B."},{"Start":"05:57.660 ","End":"06:00.275","Text":"Let me go get that formula sheet."},{"Start":"06:00.275 ","End":"06:03.860","Text":"Part B is going to be much like part A in the sense that we\u0027re going to make"},{"Start":"06:03.860 ","End":"06:08.090","Text":"a substitution and this time it\u0027s going to be 9^x."},{"Start":"06:08.090 ","End":"06:12.015","Text":"Now here, we have two subtractions in the exponent."},{"Start":"06:12.015 ","End":"06:16.970","Text":"The formula we\u0027re going to use here is this and I know we always use this one,"},{"Start":"06:16.970 ","End":"06:19.273","Text":"I\u0027ll highlight this one already."},{"Start":"06:19.273 ","End":"06:21.350","Text":"Let\u0027s see what we get,"},{"Start":"06:21.350 ","End":"06:25.410","Text":"4 is straightforward 2 over."},{"Start":"06:25.410 ","End":"06:33.410","Text":"Now, 9^x minus 1 using this formula, is 9^x/9^1."},{"Start":"06:33.410 ","End":"06:39.890","Text":"I\u0027ll just write it as 9 and here already we have 1/9."},{"Start":"06:39.890 ","End":"06:42.890","Text":"Then this equals minus"},{"Start":"06:42.890 ","End":"06:51.785","Text":"(5/9)^x over using this 9^1.5."},{"Start":"06:51.785 ","End":"06:54.920","Text":"But I don\u0027t want to waste an extra line because the power of"},{"Start":"06:54.920 ","End":"06:59.810","Text":"1/2 comes from here and it\u0027s just equal to the square root."},{"Start":"06:59.810 ","End":"07:06.400","Text":"This bit here is the square root of 9, which is 3."},{"Start":"07:06.400 ","End":"07:11.045","Text":"I looked at this as a 3 here and just save a step."},{"Start":"07:11.045 ","End":"07:16.625","Text":"Continuing, I could tidy up some more or I could make the substitution already."},{"Start":"07:16.625 ","End":"07:19.050","Text":"I\u0027ll choose tidying up."},{"Start":"07:20.900 ","End":"07:27.565","Text":"Now here, look I have 9 to fractions with a 9 on the denominator."},{"Start":"07:27.565 ","End":"07:34.010","Text":"If I multiply top and bottom here by 9 and the denominator by 9,"},{"Start":"07:34.010 ","End":"07:36.170","Text":"I should get rid of these fractions."},{"Start":"07:36.170 ","End":"07:40.370","Text":"Similarly here, if I multiply here by 3 and here by 3,"},{"Start":"07:40.370 ","End":"07:43.490","Text":"I should get rid of this denominator in the denominator."},{"Start":"07:43.490 ","End":"07:47.786","Text":"What we get is here 2 times 9, which is 18."},{"Start":"07:47.786 ","End":"07:49.700","Text":"In a normal denominator,"},{"Start":"07:49.700 ","End":"07:51.995","Text":"the 9 cancels with this 9 and this 9,"},{"Start":"07:51.995 ","End":"07:55.040","Text":"we have just 9^x minus 1."},{"Start":"07:55.040 ","End":"08:00.018","Text":"Here we have minus 5 times 3 is 15 over,"},{"Start":"08:00.018 ","End":"08:04.835","Text":"3 with the 3 cancels and I have 9^x."},{"Start":"08:04.835 ","End":"08:10.475","Text":"Now, is the point where I\u0027ll make the substitution that y=9^x."},{"Start":"08:10.475 ","End":"08:16.220","Text":"What we\u0027ll get continuing from here is 4 minus 18/y"},{"Start":"08:16.220 ","End":"08:22.890","Text":"minus 1 equals minus 15/y."},{"Start":"08:22.890 ","End":"08:26.600","Text":"This is one of those fractions equations where we have to find"},{"Start":"08:26.600 ","End":"08:30.125","Text":"the common denominator and it is clearly"},{"Start":"08:30.125 ","End":"08:34.250","Text":"y times y minus 1 or minus 1 times y. I\u0027ll write it this"},{"Start":"08:34.250 ","End":"08:38.420","Text":"way and then we have to multiply everything according to what\u0027s missing here."},{"Start":"08:38.420 ","End":"08:41.225","Text":"We\u0027re missing y minus 1."},{"Start":"08:41.225 ","End":"08:43.280","Text":"Here, we\u0027re missing y,"},{"Start":"08:43.280 ","End":"08:44.750","Text":"and here we need the whole thing,"},{"Start":"08:44.750 ","End":"08:47.568","Text":"the whole y, y minus."},{"Start":"08:47.568 ","End":"08:55.220","Text":"Multiplying out, we get this times this 4y, y minus 1."},{"Start":"08:55.220 ","End":"08:56.945","Text":"I perpetuate the numbers first."},{"Start":"08:56.945 ","End":"08:59.420","Text":"Same here, minus don\u0027t write y18,"},{"Start":"08:59.420 ","End":"09:08.495","Text":"I write 18y equals minus 15 times y minus 1."},{"Start":"09:08.495 ","End":"09:12.545","Text":"Now, let\u0027s just expand this."},{"Start":"09:12.545 ","End":"09:18.185","Text":"What I have is 4y^2 minus 4y"},{"Start":"09:18.185 ","End":"09:26.525","Text":"minus 18y equals minus 15y plus 15."},{"Start":"09:26.525 ","End":"09:32.420","Text":"Continuing over here, bring everything to the left-hand side 4y^2."},{"Start":"09:32.420 ","End":"09:37.145","Text":"Then, let\u0027s see how many y\u0027s do we have minus 4,"},{"Start":"09:37.145 ","End":"09:41.015","Text":"minus 18 plus 15."},{"Start":"09:41.015 ","End":"09:49.265","Text":"Minus 4 minus 18 is minus 22 plus 15 is still minus 7 and as for numbers,"},{"Start":"09:49.265 ","End":"09:52.040","Text":"this comes over as minus,"},{"Start":"09:52.040 ","End":"09:55.775","Text":"minus 15, then this is equal to 0."},{"Start":"09:55.775 ","End":"09:59.150","Text":"We have a quadratic equation in y."},{"Start":"09:59.150 ","End":"10:01.160","Text":"We\u0027ll use the quadratic formula,"},{"Start":"10:01.160 ","End":"10:06.590","Text":"minus b 7 plus or minus the square root of"},{"Start":"10:06.590 ","End":"10:13.130","Text":"b squared minus 4ac/2a."},{"Start":"10:13.130 ","End":"10:22.820","Text":"Let\u0027s see, 4 times 4 times 15 is 16 times 15 is 240."},{"Start":"10:22.820 ","End":"10:25.775","Text":"What we get is, the side here,"},{"Start":"10:25.775 ","End":"10:33.335","Text":"49 plus 240, that gives us 289."},{"Start":"10:33.335 ","End":"10:37.595","Text":"I happen to know that the square root of,"},{"Start":"10:37.595 ","End":"10:42.965","Text":"if I take the square root on the calculator or just happen to know that that is 17."},{"Start":"10:42.965 ","End":"10:52.520","Text":"What I have here is that y equals 7 plus or minus 17/2 times 4 is 8."},{"Start":"10:52.520 ","End":"10:58.760","Text":"Check the two solutions like the plus first 7 plus 17 is 24, 24/8 is 3."},{"Start":"10:58.760 ","End":"11:04.475","Text":"Then take the minus 7 minus 17 and minus 10 minus 10/8."},{"Start":"11:04.475 ","End":"11:07.445","Text":"Or if I cancel minus 5/4,"},{"Start":"11:07.445 ","End":"11:11.930","Text":"this is negative and experiences show that the positive will be okay,"},{"Start":"11:11.930 ","End":"11:14.255","Text":"the negative will not, will see this."},{"Start":"11:14.255 ","End":"11:18.380","Text":"If we try to go back to x using y=9^x,"},{"Start":"11:18.380 ","End":"11:20.960","Text":"just need a bit more space here."},{"Start":"11:20.960 ","End":"11:22.445","Text":"One more row should do."},{"Start":"11:22.445 ","End":"11:29.770","Text":"Then we will get that either 9^x=3,"},{"Start":"11:29.770 ","End":"11:34.900","Text":"or 9^x equals minus 5/4."},{"Start":"11:34.900 ","End":"11:37.180","Text":"Like I said, negative won\u0027t do because"},{"Start":"11:37.180 ","End":"11:40.810","Text":"a positive number 20 power is going to be positive."},{"Start":"11:40.810 ","End":"11:43.570","Text":"If positive to the positive is certainly positive,"},{"Start":"11:43.570 ","End":"11:45.160","Text":"and if it\u0027s positive to the negative,"},{"Start":"11:45.160 ","End":"11:47.380","Text":"it\u0027s one over the positive."},{"Start":"11:47.380 ","End":"11:50.980","Text":"So positive, this is immediately ruled out."},{"Start":"11:50.980 ","End":"11:57.625","Text":"All I have is 9^x=3 and I might wonder 9 to what power is 3."},{"Start":"11:57.625 ","End":"12:00.460","Text":"But I think we remember, in fact,"},{"Start":"12:00.460 ","End":"12:04.480","Text":"here it is. 9^1/2 is 3."},{"Start":"12:04.480 ","End":"12:05.830","Text":"In other words, 3 is"},{"Start":"12:05.830 ","End":"12:10.830","Text":"9^1/2 and now we have nine to the something equals nine to the something."},{"Start":"12:10.830 ","End":"12:17.765","Text":"From here, I can certainly equate the exponents and get x=1/2."},{"Start":"12:17.765 ","End":"12:24.869","Text":"That\u0027s the only answer for x and we\u0027re done with Part B also."}],"ID":8138},{"Watched":false,"Name":"Exercise 8","Duration":"13m 29s","ChapterTopicVideoID":8045,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8045.jpeg","UploadDate":"2020-09-30T14:02:21.7000000","DurationForVideoObject":"PT13M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.455","Text":"Here we have a couple of exponential equations to solve."},{"Start":"00:04.455 ","End":"00:05.670","Text":"They\u0027re similar in that,"},{"Start":"00:05.670 ","End":"00:08.445","Text":"they both will involve a substitution."},{"Start":"00:08.445 ","End":"00:16.485","Text":"From the look of it, we\u0027re going to substitute 3^x in the first and 2^x in the second."},{"Start":"00:16.485 ","End":"00:18.565","Text":"Let\u0027s get started with the first."},{"Start":"00:18.565 ","End":"00:21.730","Text":"As usual, I have a formula sheet here."},{"Start":"00:21.730 ","End":"00:24.610","Text":"Just scroll down a little bit."},{"Start":"00:25.190 ","End":"00:29.180","Text":"We see 3^x everywhere, but not quite."},{"Start":"00:29.180 ","End":"00:33.920","Text":"What I\u0027m going to do is use the formulas for the sum of"},{"Start":"00:33.920 ","End":"00:40.120","Text":"an exponent and the difference of an exponent and see if we can open this up a bit."},{"Start":"00:40.120 ","End":"00:45.030","Text":"In the first one, I have 1 minus 48 over,"},{"Start":"00:45.030 ","End":"00:52.245","Text":"now this using this is 3^x times 3^2,"},{"Start":"00:52.245 ","End":"00:54.540","Text":"sum becomes the multiplication,"},{"Start":"00:54.540 ","End":"01:01.020","Text":"minus 36 and this equals 3 to the power of minus x plus 1,"},{"Start":"01:01.020 ","End":"01:02.655","Text":"is like 1 minus x."},{"Start":"01:02.655 ","End":"01:12.405","Text":"It\u0027s 3^1 divided by 3^x all over 4 minus 3^x."},{"Start":"01:12.405 ","End":"01:16.110","Text":"At this point, I can already substitute 3^x."},{"Start":"01:16.110 ","End":"01:21.120","Text":"In fact, let\u0027s let y equals 3^x."},{"Start":"01:21.120 ","End":"01:23.910","Text":"Then what this becomes is,"},{"Start":"01:23.910 ","End":"01:32.190","Text":"1 minus 48 over 3^2 is 9 and 3^x is y."},{"Start":"01:32.190 ","End":"01:35.340","Text":"It\u0027s 9y minus 36."},{"Start":"01:35.340 ","End":"01:44.370","Text":"Here, I have 3 over y divided by 4 minus y."},{"Start":"01:44.370 ","End":"01:48.125","Text":"We have a regular equation without exponents,"},{"Start":"01:48.125 ","End":"01:50.150","Text":"but with plenty of fractions."},{"Start":"01:50.150 ","End":"01:52.124","Text":"Let\u0027s try to simplify a bit,"},{"Start":"01:52.124 ","End":"01:56.010","Text":"1 minus 48 over,"},{"Start":"01:56.010 ","End":"01:58.665","Text":"here I see that 9 goes into both of them."},{"Start":"01:58.665 ","End":"02:03.500","Text":"I\u0027ll take 9 outside the brackets and we\u0027ve got y minus 4."},{"Start":"02:03.500 ","End":"02:04.700","Text":"On the right hand side,"},{"Start":"02:04.700 ","End":"02:11.135","Text":"I can write the 3 over y as 3 over and move the y into denominator."},{"Start":"02:11.135 ","End":"02:16.115","Text":"But also look, I have y minus 4 and I have 4 minus y."},{"Start":"02:16.115 ","End":"02:21.755","Text":"One of them I could convert to the other because it\u0027s just a matter of a minus sign."},{"Start":"02:21.755 ","End":"02:27.620","Text":"Let me write the 4 minus y as y minus 4 there and to compensate,"},{"Start":"02:27.620 ","End":"02:29.570","Text":"I\u0027ll put a minus here."},{"Start":"02:29.570 ","End":"02:31.960","Text":"In general, we have the formula,"},{"Start":"02:31.960 ","End":"02:33.300","Text":"not really a formula even,"},{"Start":"02:33.300 ","End":"02:37.860","Text":"that minus a minus b is b minus a."},{"Start":"02:37.860 ","End":"02:43.890","Text":"You can reverse the order of a difference as long as you throw in an extra minus."},{"Start":"02:43.910 ","End":"02:46.665","Text":"Now, let\u0027s see anything else cancel?"},{"Start":"02:46.665 ","End":"02:48.510","Text":"Yes, 48 over 9,"},{"Start":"02:48.510 ","End":"02:51.090","Text":"they both divide by 3."},{"Start":"02:51.090 ","End":"02:57.935","Text":"To divide this by 3 and this by 3 and here I get 16 and here I get 3."},{"Start":"02:57.935 ","End":"03:05.205","Text":"What we have is 1 minus 16 over 3 times"},{"Start":"03:05.205 ","End":"03:14.040","Text":"y minus equals minus 3 over y, y minus 4."},{"Start":"03:14.040 ","End":"03:18.850","Text":"Now, we need a common denominator to get rid of the fractions."},{"Start":"03:18.850 ","End":"03:21.860","Text":"Here I have a 3, here I have a y minus 4,"},{"Start":"03:21.860 ","End":"03:23.705","Text":"also here and there is a y."},{"Start":"03:23.705 ","End":"03:30.700","Text":"All together I need 3 times y times y minus 4 to get rid of all the fractions."},{"Start":"03:30.700 ","End":"03:33.095","Text":"Let\u0027s just multiply out."},{"Start":"03:33.095 ","End":"03:35.120","Text":"I\u0027ll just mark the missing factors in each."},{"Start":"03:35.120 ","End":"03:36.660","Text":"Here, I have y minus 4,"},{"Start":"03:36.660 ","End":"03:38.285","Text":"I still need a 3."},{"Start":"03:38.285 ","End":"03:40.310","Text":"Here I have 3 y minus 4,"},{"Start":"03:40.310 ","End":"03:42.455","Text":"I still need a y."},{"Start":"03:42.455 ","End":"03:44.300","Text":"Here I need the whole thing,"},{"Start":"03:44.300 ","End":"03:48.090","Text":"3 y, y minus 4."},{"Start":"03:48.090 ","End":"03:55.410","Text":"We get 3 y times y minus 4 from here, then minus."},{"Start":"03:55.410 ","End":"03:57.585","Text":"Here I have a y,"},{"Start":"03:57.585 ","End":"03:59.465","Text":"that 3 belongs to the previous line,"},{"Start":"03:59.465 ","End":"04:03.990","Text":"y times 16, that\u0027s 16y."},{"Start":"04:03.990 ","End":"04:08.205","Text":"Here I have 3 times minus 3 is minus 9."},{"Start":"04:08.205 ","End":"04:11.130","Text":"Let\u0027s arrange this a little bit."},{"Start":"04:11.130 ","End":"04:15.270","Text":"3y^2 when I open the brackets,"},{"Start":"04:15.270 ","End":"04:19.290","Text":"minus 12y minus 16y."},{"Start":"04:19.290 ","End":"04:21.765","Text":"Let\u0027s put the 9 on the left,"},{"Start":"04:21.765 ","End":"04:25.095","Text":"that\u0027s plus 9 equals 0."},{"Start":"04:25.095 ","End":"04:29.045","Text":"Now, collect like terms and continue over here."},{"Start":"04:29.045 ","End":"04:40.050","Text":"I have 3y^2 minus 28y,"},{"Start":"04:40.250 ","End":"04:44.350","Text":"plus 9 equals 0."},{"Start":"04:44.350 ","End":"04:47.315","Text":"We\u0027ll use the quadratic formula here."},{"Start":"04:47.315 ","End":"04:58.440","Text":"We get y equals minus b plus or minus the square root of b^2, minus 4ac."},{"Start":"04:59.680 ","End":"05:02.920","Text":"This is over 2a,"},{"Start":"05:02.920 ","End":"05:05.125","Text":"which is 2 times 3."},{"Start":"05:05.125 ","End":"05:08.610","Text":"Let\u0027s see what\u0027s under the square root sign,"},{"Start":"05:08.800 ","End":"05:12.614","Text":"28^2 is 784,"},{"Start":"05:12.614 ","End":"05:20.505","Text":"4 times 3 times 9 is 12 times 9 is 108."},{"Start":"05:20.505 ","End":"05:27.960","Text":"What we get is 676."},{"Start":"05:27.960 ","End":"05:31.400","Text":"If I take the square root of this,"},{"Start":"05:31.400 ","End":"05:34.530","Text":"what I get is 26."},{"Start":"05:34.530 ","End":"05:36.195","Text":"Now back to here,"},{"Start":"05:36.195 ","End":"05:45.645","Text":"what we have is 28 plus or minus 26 over 6."},{"Start":"05:45.645 ","End":"05:53.535","Text":"Now, we split up and say 28 plus 26 is 54,"},{"Start":"05:53.535 ","End":"05:56.955","Text":"54 over 6 is 9."},{"Start":"05:56.955 ","End":"06:00.885","Text":"The other way, 28 minus 26 is 2,"},{"Start":"06:00.885 ","End":"06:04.065","Text":"2 over 6 is 1/3."},{"Start":"06:04.065 ","End":"06:07.805","Text":"These are the two solutions for y,"},{"Start":"06:07.805 ","End":"06:10.380","Text":"but y is 3^x."},{"Start":"06:10.380 ","End":"06:13.185","Text":"We have two possibilities,"},{"Start":"06:13.185 ","End":"06:20.635","Text":"3^x equals 9 or 3^x equals 1/3,"},{"Start":"06:20.635 ","End":"06:22.085","Text":"either of these is good."},{"Start":"06:22.085 ","End":"06:26.180","Text":"Let\u0027s see, the first one gives us 3^x equals,"},{"Start":"06:26.180 ","End":"06:29.975","Text":"I have to write this as 3 to the something and that\u0027s 3^2."},{"Start":"06:29.975 ","End":"06:32.630","Text":"This gives me, because I have the same base,"},{"Start":"06:32.630 ","End":"06:35.530","Text":"the exponents are equal, x equals 2."},{"Start":"06:35.530 ","End":"06:41.760","Text":"In this case, we have 3^x equals 1/3 is 3 to the minus 1."},{"Start":"06:41.760 ","End":"06:43.395","Text":"We\u0027ve seen this before."},{"Start":"06:43.395 ","End":"06:45.950","Text":"This gives us bases are equal,"},{"Start":"06:45.950 ","End":"06:50.425","Text":"compare the exponents x equals minus 1."},{"Start":"06:50.425 ","End":"06:52.460","Text":"These are the two answers for x."},{"Start":"06:52.460 ","End":"06:54.650","Text":"Both the answers should be good."},{"Start":"06:54.650 ","End":"06:57.800","Text":"It\u0027s been awhile since we verified one."},{"Start":"06:57.800 ","End":"06:59.150","Text":"Tell you what, let\u0027s take one of them."},{"Start":"06:59.150 ","End":"07:00.695","Text":"I\u0027ll take the negative 1."},{"Start":"07:00.695 ","End":"07:05.015","Text":"Let\u0027s verify this by substituting in the original equation."},{"Start":"07:05.015 ","End":"07:10.260","Text":"Here, we will put x equals minus 1."},{"Start":"07:10.260 ","End":"07:12.000","Text":"I\u0027ll just erase this."},{"Start":"07:12.000 ","End":"07:14.075","Text":"Since we\u0027re checking the equality,"},{"Start":"07:14.075 ","End":"07:16.820","Text":"let\u0027s do the left-hand side,"},{"Start":"07:16.820 ","End":"07:18.110","Text":"that\u0027s what I call LHS,"},{"Start":"07:18.110 ","End":"07:20.790","Text":"it\u0027s abbreviation for left-hand side."},{"Start":"07:20.790 ","End":"07:22.560","Text":"Separately see what it comes to,"},{"Start":"07:22.560 ","End":"07:26.930","Text":"then we\u0027ll check the right-hand side and see what it comes to."},{"Start":"07:26.930 ","End":"07:29.450","Text":"Hopefully, they come to the same thing."},{"Start":"07:29.450 ","End":"07:33.490","Text":"The left-hand side is 1 minus"},{"Start":"07:33.490 ","End":"07:40.440","Text":"48 over 3^x plus 2."},{"Start":"07:40.440 ","End":"07:41.820","Text":"If x is minus 1,"},{"Start":"07:41.820 ","End":"07:43.725","Text":"then x plus 2 is 1,"},{"Start":"07:43.725 ","End":"07:50.115","Text":"then it\u0027s 3^1 minus 36."},{"Start":"07:50.115 ","End":"07:52.250","Text":"What does that give us?"},{"Start":"07:52.250 ","End":"07:58.460","Text":"This gives us 1 minus 48 over 3 minus"},{"Start":"07:58.460 ","End":"08:07.130","Text":"36 is minus 33 and this equals 1 plus 48 over 33,"},{"Start":"08:07.130 ","End":"08:08.710","Text":"because the minuses cancel."},{"Start":"08:08.710 ","End":"08:12.095","Text":"But I\u0027ll also divide top and bottom by 3,"},{"Start":"08:12.095 ","End":"08:14.975","Text":"I\u0027ve got 16 over 11."},{"Start":"08:14.975 ","End":"08:24.090","Text":"Let\u0027s see, 16 over 11 is 1 and 5/11, 2 and 5/11."},{"Start":"08:24.090 ","End":"08:30.150","Text":"A strange number, but let\u0027s check the right-hand side now."},{"Start":"08:30.150 ","End":"08:33.195","Text":"This is, if x is minus 1,"},{"Start":"08:33.195 ","End":"08:36.800","Text":"then this is 3 to the power of plus 1, plus 1,"},{"Start":"08:36.800 ","End":"08:42.270","Text":"it\u0027s 3^2 over 4"},{"Start":"08:42.270 ","End":"08:48.670","Text":"minus 3 to the power of minus 1 is 1/3."},{"Start":"08:48.670 ","End":"08:51.575","Text":"Let\u0027s see, it\u0027s getting a bit tight here."},{"Start":"08:51.575 ","End":"08:53.905","Text":"But we can manage. Got a fraction in the bottom,"},{"Start":"08:53.905 ","End":"08:56.630","Text":"let\u0027s multiply top and bottom by 3."},{"Start":"08:56.630 ","End":"08:58.680","Text":"This is 3^2 is 9,"},{"Start":"08:58.680 ","End":"09:01.790","Text":"but I\u0027ll multiply it by 3 and on the bottom,"},{"Start":"09:01.790 ","End":"09:03.350","Text":"I\u0027ll also multiply by 3."},{"Start":"09:03.350 ","End":"09:07.370","Text":"I\u0027ve got 12 minus 1,"},{"Start":"09:07.370 ","End":"09:12.495","Text":"that gives me 27 over 11,"},{"Start":"09:12.495 ","End":"09:18.420","Text":"11 goes into 27 twice and 5 leftover."},{"Start":"09:18.420 ","End":"09:23.640","Text":"What do you know? 2 and 5/11,"},{"Start":"09:23.640 ","End":"09:26.130","Text":"those are indeed equal."},{"Start":"09:26.130 ","End":"09:29.735","Text":"This one at least is a verified solution."},{"Start":"09:29.735 ","End":"09:34.800","Text":"Hopefully, the other one is okay too and on to Part B."},{"Start":"09:34.800 ","End":"09:38.325","Text":"Let\u0027s scroll down a bit not a lot."},{"Start":"09:38.325 ","End":"09:40.800","Text":"I\u0027ll need the formula sheet."},{"Start":"09:40.800 ","End":"09:45.505","Text":"In Part B, we want to get everything in terms of 2^x."},{"Start":"09:45.505 ","End":"09:47.400","Text":"What do we do with 4^x?"},{"Start":"09:47.400 ","End":"09:52.833","Text":"It\u0027s the usual trick because 4 is 2^2,"},{"Start":"09:52.833 ","End":"09:54.805","Text":"4^x is 2^2 to the power of x."},{"Start":"09:54.805 ","End":"10:01.835","Text":"Then we can reverse the x and the 2 and say this is 2^x to the power of 2."},{"Start":"10:01.835 ","End":"10:05.845","Text":"That is because of this formula here,"},{"Start":"10:05.845 ","End":"10:08.250","Text":"2x is the same as x times 2,"},{"Start":"10:08.250 ","End":"10:12.720","Text":"but in one case I get 2^2 to the x and one case 2^x to the 2,"},{"Start":"10:12.720 ","End":"10:14.210","Text":"or you can just remember that,"},{"Start":"10:14.210 ","End":"10:15.920","Text":"you can reverse these two."},{"Start":"10:15.920 ","End":"10:22.340","Text":"What we get here is 5 times 2^x minus 10 over"},{"Start":"10:22.340 ","End":"10:29.185","Text":"2^x squared minus 2^x minus 2 equals 1."},{"Start":"10:29.185 ","End":"10:31.785","Text":"Now everything is in terms of 2^x."},{"Start":"10:31.785 ","End":"10:36.785","Text":"Now we can substitute y equals 2^x."},{"Start":"10:36.785 ","End":"10:44.329","Text":"Then continuing, we have 5y minus 10 over"},{"Start":"10:44.329 ","End":"10:52.575","Text":"y^2 minus y minus 2 equals 1."},{"Start":"10:52.575 ","End":"10:56.930","Text":"Let\u0027s just multiply both sides by the denominator here."},{"Start":"10:56.930 ","End":"10:58.820","Text":"I\u0027ll get that this equals this."},{"Start":"10:58.820 ","End":"11:00.320","Text":"Another way of looking at it is that,"},{"Start":"11:00.320 ","End":"11:01.910","Text":"if a over b is one,"},{"Start":"11:01.910 ","End":"11:03.110","Text":"then a equals b,"},{"Start":"11:03.110 ","End":"11:09.815","Text":"either way 5y minus 10 equals y^2,"},{"Start":"11:09.815 ","End":"11:13.760","Text":"minus y minus 2."},{"Start":"11:13.760 ","End":"11:15.890","Text":"That\u0027s a quadratic equation."},{"Start":"11:15.890 ","End":"11:17.555","Text":"You just have to arrange it a bit."},{"Start":"11:17.555 ","End":"11:20.835","Text":"Let\u0027s move everything to the left."},{"Start":"11:20.835 ","End":"11:23.929","Text":"We have minus y^2 from here."},{"Start":"11:23.929 ","End":"11:27.380","Text":"Let us see, the y we have here and here, plus 5,"},{"Start":"11:27.380 ","End":"11:35.024","Text":"plus 1, 6y and minus 10 plus 2 is minus 8 equals 0."},{"Start":"11:35.024 ","End":"11:38.495","Text":"I really prefer to have the a as positive."},{"Start":"11:38.495 ","End":"11:40.850","Text":"Let\u0027s just multiply everything by minus."},{"Start":"11:40.850 ","End":"11:49.160","Text":"We\u0027ve got y^2 minus 6 plus 8 equals 0. Quadratic equation in y."},{"Start":"11:49.160 ","End":"11:53.965","Text":"Formula, y equals minus b,"},{"Start":"11:53.965 ","End":"12:03.920","Text":"plus or minus the square root of b^2 minus 4ac all over 2a."},{"Start":"12:03.920 ","End":"12:07.735","Text":"Let\u0027s see, what\u0027s under the square root sign?"},{"Start":"12:07.735 ","End":"12:13.365","Text":"6^2 is 36, 4 times 8 is 32."},{"Start":"12:13.365 ","End":"12:17.715","Text":"Let\u0027s see, we had 36 minus 32,"},{"Start":"12:17.715 ","End":"12:22.965","Text":"that\u0027s 4 and the square root of 4 we know is 2."},{"Start":"12:22.965 ","End":"12:31.880","Text":"This comes out as 6 plus or minus 2 and 2 times 1 is 2."},{"Start":"12:31.880 ","End":"12:34.145","Text":"What are our possibilities?"},{"Start":"12:34.145 ","End":"12:36.665","Text":"6 plus 2 is 8."},{"Start":"12:36.665 ","End":"12:41.940","Text":"8 over 2 is 4 and 6 minus 2 is 4,"},{"Start":"12:41.940 ","End":"12:43.860","Text":"4 over 2 is 2."},{"Start":"12:43.860 ","End":"12:46.920","Text":"These two answers are for y not for x."},{"Start":"12:46.920 ","End":"12:53.030","Text":"We still need to go back to this formula, the substitution really."},{"Start":"12:53.030 ","End":"13:01.330","Text":"We get that either 2^x is 4 or 2^x is 2."},{"Start":"13:01.330 ","End":"13:05.000","Text":"In each case, I\u0027d like to write them as 2 to the power of something,"},{"Start":"13:05.000 ","End":"13:09.145","Text":"4 is 2^2, 2 is 2^1."},{"Start":"13:09.145 ","End":"13:12.300","Text":"If I have the same base 2,"},{"Start":"13:12.300 ","End":"13:14.379","Text":"then the exponents are equal,"},{"Start":"13:14.379 ","End":"13:16.335","Text":"so x equals 2 and the same thing here."},{"Start":"13:16.335 ","End":"13:18.640","Text":"These two and these two are the same base,"},{"Start":"13:18.640 ","End":"13:20.710","Text":"so the exponents are equal."},{"Start":"13:20.710 ","End":"13:24.710","Text":"This time, these are the solutions for x."},{"Start":"13:24.710 ","End":"13:26.735","Text":"We\u0027re done with Part B."},{"Start":"13:26.735 ","End":"13:29.670","Text":"Let\u0027s see under the exercise."}],"ID":8139},{"Watched":false,"Name":"Exercise 9","Duration":"14m 18s","ChapterTopicVideoID":8046,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8046.jpeg","UploadDate":"2020-09-30T14:08:54.3570000","DurationForVideoObject":"PT14M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"Here we have another couple of equations."},{"Start":"00:03.090 ","End":"00:04.650","Text":"They\u0027re both exponential,"},{"Start":"00:04.650 ","End":"00:07.905","Text":"and in both cases we\u0027re going to be using substitution."},{"Start":"00:07.905 ","End":"00:11.924","Text":"They get a bit more delicate as you get more advanced."},{"Start":"00:11.924 ","End":"00:13.650","Text":"The main question is,"},{"Start":"00:13.650 ","End":"00:15.539","Text":"what are you going to substitute?"},{"Start":"00:15.539 ","End":"00:18.150","Text":"To do a quick analysis of what\u0027s going on here,"},{"Start":"00:18.150 ","End":"00:19.785","Text":"let\u0027s start with a."},{"Start":"00:19.785 ","End":"00:22.665","Text":"As usual, we have the formula sheet in case we need it."},{"Start":"00:22.665 ","End":"00:28.718","Text":"I\u0027m looking here and I see I have 7^2 square root of x,"},{"Start":"00:28.718 ","End":"00:32.100","Text":"and the same twice square root of x appears here."},{"Start":"00:32.100 ","End":"00:33.645","Text":"Because it appears twice,"},{"Start":"00:33.645 ","End":"00:39.635","Text":"I don\u0027t see any need to decompose it into two and root x separately."},{"Start":"00:39.635 ","End":"00:44.030","Text":"My first intuition tells me to take the whole 7 to the power of twice"},{"Start":"00:44.030 ","End":"00:48.710","Text":"root x as my variable to as the expression to substitute."},{"Start":"00:48.710 ","End":"00:51.080","Text":"Then I\u0027m going to ask yourself,"},{"Start":"00:51.080 ","End":"00:53.945","Text":"well, how will I get around the 49?"},{"Start":"00:53.945 ","End":"00:56.885","Text":"49 will just be 7^2."},{"Start":"00:56.885 ","End":"01:00.455","Text":"It\u0027ll just be this thing squared."},{"Start":"01:00.455 ","End":"01:03.325","Text":"Let\u0027s just write something down and I\u0027ll show you."},{"Start":"01:03.325 ","End":"01:07.700","Text":"Unlike previous exercises, I\u0027m going to declare my intent upfront."},{"Start":"01:07.700 ","End":"01:12.650","Text":"I\u0027m going to want to substitute Y=7^2 root x."},{"Start":"01:12.650 ","End":"01:15.880","Text":"Only I\u0027m not quite ready yet because of this thing here."},{"Start":"01:15.880 ","End":"01:19.085","Text":"What I\u0027m going to do is my usual trick,"},{"Start":"01:19.085 ","End":"01:21.080","Text":"because it\u0027s 49,"},{"Start":"01:21.080 ","End":"01:26.495","Text":"I\u0027m going to write it as 7^2 root x all squared."},{"Start":"01:26.495 ","End":"01:27.980","Text":"Now how do I do this?"},{"Start":"01:27.980 ","End":"01:29.660","Text":"Because 49 is 7^2,"},{"Start":"01:29.660 ","End":"01:31.760","Text":"how did the certainly get here?"},{"Start":"01:31.760 ","End":"01:33.995","Text":"Again, I\u0027ll show you the usual trick."},{"Start":"01:33.995 ","End":"01:35.705","Text":"I\u0027ll do it at the side here,"},{"Start":"01:35.705 ","End":"01:40.497","Text":"49^2 root x is 7 squared,"},{"Start":"01:40.497 ","End":"01:44.450","Text":"that\u0027s just the 49^2 root x."},{"Start":"01:44.450 ","End":"01:47.405","Text":"Now, I mentioned before, no,"},{"Start":"01:47.405 ","End":"01:51.350","Text":"remind you that we can reverse the order of these two."},{"Start":"01:51.350 ","End":"01:53.180","Text":"Why can we reverse the order?"},{"Start":"01:53.180 ","End":"01:55.535","Text":"It all comes down to this formula."},{"Start":"01:55.535 ","End":"01:59.555","Text":"I could actually rewrite this formula and say that this is also"},{"Start":"01:59.555 ","End":"02:04.067","Text":"equal to a^n to the power of"},{"Start":"02:04.067 ","End":"02:13.250","Text":"m. The reason it does not matter which way around,"},{"Start":"02:13.250 ","End":"02:14.330","Text":"is because m times m equals m times m. That didn\u0027t come out very clearly,"},{"Start":"02:14.330 ","End":"02:15.380","Text":"I\u0027ll just erase it."},{"Start":"02:15.380 ","End":"02:16.600","Text":"Which will just tell you what I mean,"},{"Start":"02:16.600 ","End":"02:19.700","Text":"m times n is n times m and if I just happen to reverse it,"},{"Start":"02:19.700 ","End":"02:21.015","Text":"I would get the other thing."},{"Start":"02:21.015 ","End":"02:25.445","Text":"I can always switch the order and if I switch the order,"},{"Start":"02:25.445 ","End":"02:28.775","Text":"then I get 7^2 root x,"},{"Start":"02:28.775 ","End":"02:33.395","Text":"this bottom on the inside and the other 2 on the outside and that\u0027s what I did here."},{"Start":"02:33.395 ","End":"02:43.550","Text":"Now, just continue minus 8 times 7^2 root x plus 7 equals 0."},{"Start":"02:43.550 ","End":"02:48.170","Text":"Now, everywhere I have 7^2 root x. I have it here,"},{"Start":"02:48.170 ","End":"02:51.980","Text":"and I have it here this is what I want to substitute,"},{"Start":"02:51.980 ","End":"02:53.600","Text":"and it works out nicely."},{"Start":"02:53.600 ","End":"02:58.280","Text":"Because now, this is what I marked in this color is just y. I have y"},{"Start":"02:58.280 ","End":"03:04.505","Text":"squared minus 8 times y plus 7 equals 0."},{"Start":"03:04.505 ","End":"03:08.320","Text":"This is a simple quadratic equation in y."},{"Start":"03:08.320 ","End":"03:11.599","Text":"Let\u0027s start solving it using the formula."},{"Start":"03:11.599 ","End":"03:21.900","Text":"We get y equals minus b plus or minus the square root of b^2 minus 4ac,"},{"Start":"03:21.900 ","End":"03:25.720","Text":"all this is over 2a."},{"Start":"03:25.720 ","End":"03:30.660","Text":"As usual, what interests me is what\u0027s under the square root."},{"Start":"03:30.660 ","End":"03:34.260","Text":"Let\u0027s see, 8^2 is 64,"},{"Start":"03:34.260 ","End":"03:38.415","Text":"4 times 7 is 28,"},{"Start":"03:38.415 ","End":"03:44.498","Text":"64 minus 28 is 36,"},{"Start":"03:44.498 ","End":"03:47.940","Text":"and 36 is 6^2."},{"Start":"03:47.940 ","End":"03:50.310","Text":"The square root will come out to be 6."},{"Start":"03:50.310 ","End":"03:57.270","Text":"What I get is 8 plus or minus 6 over 2 times 1 is 2."},{"Start":"03:57.270 ","End":"04:00.390","Text":"Now, I take plus or minus respectively,"},{"Start":"04:00.390 ","End":"04:03.690","Text":"8 plus 6 is 14,"},{"Start":"04:03.690 ","End":"04:05.970","Text":"14 over 2 is 7,"},{"Start":"04:05.970 ","End":"04:08.910","Text":"8 minus 6 is 2,"},{"Start":"04:08.910 ","End":"04:10.810","Text":"2 over 2 is 1."},{"Start":"04:10.810 ","End":"04:15.260","Text":"These are my answers for y, 7 or 1."},{"Start":"04:15.260 ","End":"04:17.110","Text":"Now, when I get back to x,"},{"Start":"04:17.110 ","End":"04:18.920","Text":"we remember where we came from,"},{"Start":"04:18.920 ","End":"04:28.305","Text":"y is 7^2 root x. I have 7^2 root x is equal to 7,"},{"Start":"04:28.305 ","End":"04:32.830","Text":"or 7^2 root x is equal to 1."},{"Start":"04:32.830 ","End":"04:35.675","Text":"I\u0027m just putting y what it was from here."},{"Start":"04:35.675 ","End":"04:38.145","Text":"Now, I have 7 to the something else,"},{"Start":"04:38.145 ","End":"04:39.230","Text":"1, 7 to the something,"},{"Start":"04:39.230 ","End":"04:44.300","Text":"7 I\u0027ll rewrite as 7^1 and 1 I\u0027ll write as 7^0."},{"Start":"04:44.300 ","End":"04:45.815","Text":"In case you\u0027ve forgotten about that."},{"Start":"04:45.815 ","End":"04:49.115","Text":"That\u0027s this formula here with a equal 7."},{"Start":"04:49.115 ","End":"04:53.630","Text":"We\u0027re at the situation now where we have the same base."},{"Start":"04:53.630 ","End":"04:59.730","Text":"We compare the exponents that gives 2 root x equals 1"},{"Start":"04:59.730 ","End":"05:07.829","Text":"and then that means that root x equals 1/2."},{"Start":"05:07.829 ","End":"05:10.430","Text":"Now how do I get from root x to x?"},{"Start":"05:10.430 ","End":"05:12.650","Text":"I just square both sides."},{"Start":"05:12.650 ","End":"05:16.543","Text":"That\u0027s okay as long as this is a positive number,"},{"Start":"05:16.543 ","End":"05:20.780","Text":"so we get that from here if we square both sides."},{"Start":"05:20.780 ","End":"05:25.110","Text":"I\u0027ll just write that I\u0027m squaring both sides are taken to the power of 2,"},{"Start":"05:25.110 ","End":"05:27.955","Text":"I get that x equals a 1/2 squared,"},{"Start":"05:27.955 ","End":"05:29.635","Text":"which is 1/4,"},{"Start":"05:29.635 ","End":"05:32.090","Text":"1 times 1 over 2 times 2."},{"Start":"05:32.090 ","End":"05:34.490","Text":"That takes care of this possibility."},{"Start":"05:34.490 ","End":"05:36.005","Text":"Now the other possibility,"},{"Start":"05:36.005 ","End":"05:39.620","Text":"once again, I have the base and the base."},{"Start":"05:39.620 ","End":"05:41.815","Text":"I compare the exponents."},{"Start":"05:41.815 ","End":"05:46.275","Text":"This time, 2 root x from here equals 0 from here,"},{"Start":"05:46.275 ","End":"05:50.560","Text":"divide both sides by 2 root x equals 0."},{"Start":"05:50.560 ","End":"05:52.764","Text":"The square root of something is 0."},{"Start":"05:52.764 ","End":"05:55.330","Text":"Then it also must be 0,"},{"Start":"05:55.330 ","End":"05:58.915","Text":"or if you prefer 0^2 is still 0."},{"Start":"05:58.915 ","End":"06:02.689","Text":"The two solutions for x,"},{"Start":"06:02.790 ","End":"06:07.915","Text":"1/4 or 0, and that solves part a."},{"Start":"06:07.915 ","End":"06:09.790","Text":"Let\u0027s do a quick check on one of them."},{"Start":"06:09.790 ","End":"06:11.710","Text":"I\u0027ll check on the easier one, the 0."},{"Start":"06:11.710 ","End":"06:14.725","Text":"Let\u0027s see if this works in the original equation."},{"Start":"06:14.725 ","End":"06:18.205","Text":"I\u0027m going to put x equals 0 in here."},{"Start":"06:18.205 ","End":"06:21.455","Text":"I\u0027m just going to do it on the left-hand side and see what I get."},{"Start":"06:21.455 ","End":"06:28.630","Text":"If x is 0, I have 49^2 root 0 minus"},{"Start":"06:28.630 ","End":"06:36.525","Text":"8 times 7^2 root 0 plus 7."},{"Start":"06:36.525 ","End":"06:37.770","Text":"That\u0027s the left-hand side."},{"Start":"06:37.770 ","End":"06:41.285","Text":"Let\u0027s develop it and see what we get and hopefully we\u0027ll end up with 0."},{"Start":"06:41.285 ","End":"06:43.430","Text":"Well, root 0 is 0,"},{"Start":"06:43.430 ","End":"06:44.840","Text":"twice 0 is 0."},{"Start":"06:44.840 ","End":"06:48.235","Text":"Anything to the power of 0 is 1."},{"Start":"06:48.235 ","End":"06:52.460","Text":"This is just 1 and then 7^0,"},{"Start":"06:52.460 ","End":"06:54.200","Text":"same thing here is 1,"},{"Start":"06:54.200 ","End":"06:59.680","Text":"so it\u0027s minus 8 times 1 plus 7."},{"Start":"06:59.680 ","End":"07:02.635","Text":"We have 1 plus 7 minus 8,"},{"Start":"07:02.635 ","End":"07:04.250","Text":"which is indeed 0,"},{"Start":"07:04.250 ","End":"07:06.269","Text":"which is the right-hand side,"},{"Start":"07:06.269 ","End":"07:10.400","Text":"so we can say that x=0 is a verified solution if you like,"},{"Start":"07:10.400 ","End":"07:13.580","Text":"we\u0027ve checked it and if you have time,"},{"Start":"07:13.580 ","End":"07:16.355","Text":"I recommend checking the other one also."},{"Start":"07:16.355 ","End":"07:19.760","Text":"Let\u0027s go on to Part B."},{"Start":"07:19.760 ","End":"07:26.255","Text":"Just scroll down a bit till I get to Part B and I\u0027m going to need my formula sheet."},{"Start":"07:26.255 ","End":"07:33.260","Text":"Here we are and now I\u0027m looking at this and this and seeing that I have 2 and I have 4,"},{"Start":"07:33.260 ","End":"07:36.575","Text":"so I\u0027m probably going to want to substitute 2 to the power of something."},{"Start":"07:36.575 ","End":"07:39.890","Text":"Exactly what, whether it\u0027s 2 to the x or 2 to the 2x,"},{"Start":"07:39.890 ","End":"07:42.935","Text":"it\u0027s hard to say until We develop this a bit."},{"Start":"07:42.935 ","End":"07:48.785","Text":"What I suggest is simplifying the denominator here and get it to be in terms of 2."},{"Start":"07:48.785 ","End":"07:50.195","Text":"I\u0027ll do this at the side,"},{"Start":"07:50.195 ","End":"07:51.770","Text":"I\u0027m taking the denominator here,"},{"Start":"07:51.770 ","End":"07:57.215","Text":"4^0.5x minus 1,"},{"Start":"07:57.215 ","End":"08:00.283","Text":"I can better write this 4 as 2^2,"},{"Start":"08:00.283 ","End":"08:08.390","Text":"(2^2) to the power of 0.5x minus 1."},{"Start":"08:08.390 ","End":"08:13.145","Text":"Now I\u0027ll need the formula here but from right to left."},{"Start":"08:13.145 ","End":"08:17.240","Text":"I\u0027m going to use m equals 2 and n is going to be that thing here,"},{"Start":"08:17.240 ","End":"08:19.040","Text":"just write the equals."},{"Start":"08:19.040 ","End":"08:22.580","Text":"What I get is 2 to the power of this times this."},{"Start":"08:22.580 ","End":"08:24.500","Text":"Let us multiply it out straightaway,"},{"Start":"08:24.500 ","End":"08:29.750","Text":"2 times 0.5x is just x because 2 times 0.5 is 1,"},{"Start":"08:29.750 ","End":"08:39.015","Text":"just like 2 times a 1/2 and minus 2 and that\u0027s what I have for the denominator here."},{"Start":"08:39.015 ","End":"08:43.270","Text":"Let\u0027s rewrite this. Now,"},{"Start":"08:43.270 ","End":"08:46.085","Text":"we also have a problem because there\u0027s a square root here."},{"Start":"08:46.085 ","End":"08:51.395","Text":"I\u0027m also going to rewrite this expression here, another side exercise."},{"Start":"08:51.395 ","End":"08:55.220","Text":"This time, we\u0027ll take the square root is different color for"},{"Start":"08:55.220 ","End":"09:02.315","Text":"separate exercises of 2 to the 2x plus 2 and see what this equals."},{"Start":"09:02.315 ","End":"09:04.580","Text":"This time I\u0027m going to use this formula."},{"Start":"09:04.580 ","End":"09:08.810","Text":"It tells me that the square root of something is that thing to the power of 1/2."},{"Start":"09:08.810 ","End":"09:16.235","Text":"What We have here is 2 to the power of 2x plus 2 to the power of 1/2."},{"Start":"09:16.235 ","End":"09:21.050","Text":"Once again, using this formula from right to left,"},{"Start":"09:21.050 ","End":"09:24.350","Text":"what it tells me to do is to multiply the 2 exponents."},{"Start":"09:24.350 ","End":"09:28.040","Text":"This becomes 2 to the power of this times this."},{"Start":"09:28.040 ","End":"09:29.360","Text":"I can multiply in any order,"},{"Start":"09:29.360 ","End":"09:34.215","Text":"I just prefer to put the 1/2 first, 2x plus 2."},{"Start":"09:34.215 ","End":"09:37.105","Text":"Now I\u0027ll just multiply out this here,"},{"Start":"09:37.105 ","End":"09:45.350","Text":"it\u0027s 2 to the power of 1/2 times 2x is x and 1/2 times 2 is 1, I get this."},{"Start":"09:45.350 ","End":"09:50.975","Text":"As for both of these side exercises back into the main exercise and what I have is,"},{"Start":"09:50.975 ","End":"09:56.830","Text":"this was this color that\u0027s 2 to the x plus 1, plus 1 over."},{"Start":"09:56.830 ","End":"09:58.837","Text":"This was the one I did in this color,"},{"Start":"09:58.837 ","End":"10:03.720","Text":"that\u0027s 2 to the x minus 2 and this equals 9."},{"Start":"10:03.720 ","End":"10:08.450","Text":"It seems clear to me that what I want to do is isolate 2 to the power of x."},{"Start":"10:08.450 ","End":"10:11.900","Text":"Here, I have x plus 1 and here I have x minus 2."},{"Start":"10:11.900 ","End":"10:16.160","Text":"For the plus, I could use this formula and for the minus,"},{"Start":"10:16.160 ","End":"10:18.425","Text":"I can use this formula."},{"Start":"10:18.425 ","End":"10:20.435","Text":"Let\u0027s do that here."},{"Start":"10:20.435 ","End":"10:26.105","Text":"This will be from the first formula 2^x times 2^1."},{"Start":"10:26.105 ","End":"10:29.525","Text":"I won\u0027t write 2 to the 1 because it\u0027s just 2."},{"Start":"10:29.525 ","End":"10:31.715","Text":"Here, we\u0027ll have 1 over."},{"Start":"10:31.715 ","End":"10:37.220","Text":"Now, 2 to the x minus 2 will be 2 to the x over 2 to the 2."},{"Start":"10:37.220 ","End":"10:44.285","Text":"That\u0027s 2 to the x over 2 to the 2 and all this is equal to 9."},{"Start":"10:44.285 ","End":"10:46.280","Text":"When you divide by a fraction,"},{"Start":"10:46.280 ","End":"10:48.695","Text":"1 over a fraction is the inverse fraction."},{"Start":"10:48.695 ","End":"10:50.659","Text":"Let me just do the middle one first."},{"Start":"10:50.659 ","End":"10:55.400","Text":"The middle one becomes 2 to the 2 over 2 to the x."},{"Start":"10:55.400 ","End":"11:00.830","Text":"Reminding you 1 over a over b is b over a."},{"Start":"11:00.830 ","End":"11:06.230","Text":"Dividing by a fraction is multiplying by the reciprocal fractions."},{"Start":"11:06.230 ","End":"11:08.480","Text":"Here I have, I\u0027ll put the 2 first,"},{"Start":"11:08.480 ","End":"11:12.785","Text":"2 times 2 to the x and all this is equal to 9."},{"Start":"11:12.785 ","End":"11:14.600","Text":"I think I\u0027m about ready to substitute,"},{"Start":"11:14.600 ","End":"11:16.190","Text":"because I have 2 to the x here,"},{"Start":"11:16.190 ","End":"11:19.100","Text":"and I have 2 to the x here."},{"Start":"11:19.100 ","End":"11:20.870","Text":"These signs might be confusing,"},{"Start":"11:20.870 ","End":"11:22.475","Text":"but see what I mean."},{"Start":"11:22.475 ","End":"11:27.275","Text":"Now, I\u0027m going to write my intention is to substitute y equals 2 to the x."},{"Start":"11:27.275 ","End":"11:30.890","Text":"Perhaps I\u0027ll circle them here and here,"},{"Start":"11:30.890 ","End":"11:41.845","Text":"2y plus 2^2 is 4,4 over y equals 9."},{"Start":"11:41.845 ","End":"11:44.110","Text":"That\u0027s a straightforward fraction equation,"},{"Start":"11:44.110 ","End":"11:45.520","Text":"but there\u0027s no exponents."},{"Start":"11:45.520 ","End":"11:50.760","Text":"Just multiply both sides by y and that\u0027ll get rid of the fraction."},{"Start":"11:50.760 ","End":"11:54.500","Text":"We\u0027ll get 2y times y is 2y^2."},{"Start":"11:54.500 ","End":"11:58.640","Text":"Here, the y just cancels and I\u0027m left with 4 and here I get"},{"Start":"11:58.640 ","End":"12:03.425","Text":"9 times y. I bring everything to the left and put it in the correct order,"},{"Start":"12:03.425 ","End":"12:06.590","Text":"it\u0027s 2y squared minus 9y,"},{"Start":"12:06.590 ","End":"12:10.745","Text":"because it came from the other side and plus 4 equals 0."},{"Start":"12:10.745 ","End":"12:12.815","Text":"Quadratic equation in y,"},{"Start":"12:12.815 ","End":"12:15.605","Text":"which seems to be often the case in these exercises."},{"Start":"12:15.605 ","End":"12:21.980","Text":"I\u0027ll do it over here and I get that y equals minus b plus"},{"Start":"12:21.980 ","End":"12:28.580","Text":"9 plus or minus the square root of b squared minus 4 times a,"},{"Start":"12:28.580 ","End":"12:30.320","Text":"which is 2 times c,"},{"Start":"12:30.320 ","End":"12:35.840","Text":"which is 4 all over 2a which is 2 times 2."},{"Start":"12:35.840 ","End":"12:38.690","Text":"Let\u0027s see, what\u0027s under the square root sign?"},{"Start":"12:38.690 ","End":"12:42.170","Text":"9 squared is 81 minus,"},{"Start":"12:42.170 ","End":"12:47.280","Text":"2 times 4 times 2 is 32."},{"Start":"12:47.530 ","End":"12:49.790","Text":"What am I left with?"},{"Start":"12:49.790 ","End":"12:55.145","Text":"That\u0027s 49, and 49 I know that that\u0027s 7 squared,"},{"Start":"12:55.145 ","End":"12:59.915","Text":"so when I take the square root of 49, it\u0027ll be 7."},{"Start":"12:59.915 ","End":"13:05.315","Text":"I get 9 plus or minus 7 square root of 49,"},{"Start":"13:05.315 ","End":"13:07.850","Text":"over 2 times 2 is 4."},{"Start":"13:07.850 ","End":"13:11.000","Text":"Now, I get the two possibilities from the plus minus."},{"Start":"13:11.000 ","End":"13:15.305","Text":"If I take plus, 9 plus 7 is 16 over 4,"},{"Start":"13:15.305 ","End":"13:22.385","Text":"is 4 and 9 minus 7 is 2 over 4, which is 1/2."},{"Start":"13:22.385 ","End":"13:25.850","Text":"These are the solutions for y,"},{"Start":"13:25.850 ","End":"13:29.525","Text":"which is 2 to the x. I\u0027ll continue over here."},{"Start":"13:29.525 ","End":"13:31.145","Text":"I have two possibilities,"},{"Start":"13:31.145 ","End":"13:34.010","Text":"2^x is 4,"},{"Start":"13:34.010 ","End":"13:39.230","Text":"or 2^x is 1/2."},{"Start":"13:39.230 ","End":"13:42.650","Text":"Now, way to do this is just to write 4 as 2 to the power of something,"},{"Start":"13:42.650 ","End":"13:44.240","Text":"2 to the power of what is 4?"},{"Start":"13:44.240 ","End":"13:49.280","Text":"We know that 2 to the power of 2 and 1/2 is 2 to the power of minus 1."},{"Start":"13:49.280 ","End":"13:50.825","Text":"We\u0027ve seen this several times."},{"Start":"13:50.825 ","End":"13:55.970","Text":"I refer you to this formula with n equals 1,"},{"Start":"13:55.970 ","End":"14:00.080","Text":"and now 2 the x equals 2 to the 2 same base."},{"Start":"14:00.080 ","End":"14:07.400","Text":"This gives me that the exponents are equal x equals 2 and this gives me again,"},{"Start":"14:07.400 ","End":"14:12.290","Text":"the same base 2 exponents are equal x equals minus 1."},{"Start":"14:12.290 ","End":"14:14.660","Text":"Just highlight those two answers."},{"Start":"14:14.660 ","End":"14:18.750","Text":"This is the answer of Part B and so We are done."}],"ID":8140},{"Watched":false,"Name":"Exercise 10","Duration":"6m 55s","ChapterTopicVideoID":8047,"CourseChapterTopicPlaylistID":56157,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8047.jpeg","UploadDate":"2020-09-30T14:11:46.3800000","DurationForVideoObject":"PT6M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.235","Text":"In this exercise, we have another pair of exponential equations to solve,"},{"Start":"00:05.235 ","End":"00:07.080","Text":"and we\u0027ll begin with the first."},{"Start":"00:07.080 ","End":"00:08.880","Text":"I\u0027ll assume you know what e is,"},{"Start":"00:08.880 ","End":"00:09.960","Text":"but even if you don\u0027t,"},{"Start":"00:09.960 ","End":"00:12.645","Text":"you\u0027re welcome to stay it will make perfect sense."},{"Start":"00:12.645 ","End":"00:15.075","Text":"E is just some number between 2 and 3."},{"Start":"00:15.075 ","End":"00:17.265","Text":"That\u0027s all you really need to know."},{"Start":"00:17.265 ","End":"00:20.220","Text":"I see here e^x,"},{"Start":"00:20.220 ","End":"00:21.810","Text":"I see here e^2x."},{"Start":"00:21.810 ","End":"00:24.000","Text":"I already know the relationship between them because"},{"Start":"00:24.000 ","End":"00:26.475","Text":"we\u0027ve done this with other numbers many times."},{"Start":"00:26.475 ","End":"00:28.545","Text":"But let me show you what the side,"},{"Start":"00:28.545 ","End":"00:33.215","Text":"e^2x, I want to use this formula on it."},{"Start":"00:33.215 ","End":"00:34.670","Text":"If I just did it as is,"},{"Start":"00:34.670 ","End":"00:38.060","Text":"it would tell me that this is e^2^ x."},{"Start":"00:38.060 ","End":"00:44.090","Text":"But if I reverse the order and think of this as e^x times 2,"},{"Start":"00:44.090 ","End":"00:50.305","Text":"then the formula will tell me that it\u0027s (e^x)^2, the standard trick."},{"Start":"00:50.305 ","End":"00:58.910","Text":"If I write this now as e^x squared minus 2e^x plus 1 equals 0,"},{"Start":"00:58.910 ","End":"01:03.390","Text":"you can see that e^x is just begging to be substituted."},{"Start":"01:03.390 ","End":"01:05.915","Text":"I can say everything is involved e^x,"},{"Start":"01:05.915 ","End":"01:10.220","Text":"let\u0027s call that y. Y equals e^x."},{"Start":"01:10.220 ","End":"01:19.325","Text":"Then this thing becomes y^2 minus 2y plus 1= 0."},{"Start":"01:19.325 ","End":"01:22.685","Text":"Let\u0027s solve this equation for y."},{"Start":"01:22.685 ","End":"01:25.714","Text":"Let\u0027s use the formula that there are other ways."},{"Start":"01:25.714 ","End":"01:28.730","Text":"I will say that y equals minus b,"},{"Start":"01:28.730 ","End":"01:30.080","Text":"which is 2,"},{"Start":"01:30.080 ","End":"01:35.240","Text":"plus or minus the square root of b^2, which is 4,"},{"Start":"01:35.240 ","End":"01:39.200","Text":"minus 4ac, 4 times 1 times 1,"},{"Start":"01:39.200 ","End":"01:41.110","Text":"all this over 2."},{"Start":"01:41.110 ","End":"01:47.250","Text":"Now what I get is 2 plus or minus 0. Why 0?"},{"Start":"01:47.250 ","End":"01:51.560","Text":"Because 4 minus 4 times 1 times 1 is 4 minus 4 is 0,"},{"Start":"01:51.560 ","End":"01:54.220","Text":"and the square root of 0 is 0/2."},{"Start":"01:54.220 ","End":"01:56.360","Text":"When we have plus or minus 0."},{"Start":"01:56.360 ","End":"01:58.460","Text":"We don\u0027t get two solutions, we only get one."},{"Start":"01:58.460 ","End":"02:00.080","Text":"We can just throw out the 0."},{"Start":"02:00.080 ","End":"02:03.205","Text":"This is 2/ 2, which is 1."},{"Start":"02:03.205 ","End":"02:05.240","Text":"Now that we have that y is 1,"},{"Start":"02:05.240 ","End":"02:10.940","Text":"we can go back here and say that e^x is 1."},{"Start":"02:10.940 ","End":"02:14.705","Text":"Now 1 is e^0,"},{"Start":"02:14.705 ","End":"02:16.085","Text":"because of this formula,"},{"Start":"02:16.085 ","End":"02:18.050","Text":"anything to the 0 is 1."},{"Start":"02:18.050 ","End":"02:21.245","Text":"So e^x equals e^0."},{"Start":"02:21.245 ","End":"02:24.290","Text":"Our basic principle in these exponential equations,"},{"Start":"02:24.290 ","End":"02:26.020","Text":"if we have the same base,"},{"Start":"02:26.020 ","End":"02:27.510","Text":"e and e are the same,"},{"Start":"02:27.510 ","End":"02:30.695","Text":"so the exponents have to be the same also."},{"Start":"02:30.695 ","End":"02:33.145","Text":"X has to equal 0,"},{"Start":"02:33.145 ","End":"02:36.435","Text":"and that\u0027s our answer."},{"Start":"02:36.435 ","End":"02:39.685","Text":"Onto part b."},{"Start":"02:39.685 ","End":"02:42.350","Text":"Just to scroll down to it."},{"Start":"02:42.350 ","End":"02:46.250","Text":"Here we are, and I\u0027ll go and stretch that formula table."},{"Start":"02:46.250 ","End":"02:48.380","Text":"There it is. Let\u0027s get started."},{"Start":"02:48.380 ","End":"02:49.910","Text":"Now in part a,"},{"Start":"02:49.910 ","End":"02:51.605","Text":"we had e^2x,"},{"Start":"02:51.605 ","End":"02:53.540","Text":"so I\u0027m not going to do it again."},{"Start":"02:53.540 ","End":"02:56.135","Text":"We figured was (e^x)^2."},{"Start":"02:56.135 ","End":"02:58.910","Text":"Once again, I want to get everything in terms of e^ x."},{"Start":"02:58.910 ","End":"03:04.460","Text":"Here I see opportunity to use this formula because"},{"Start":"03:04.460 ","End":"03:10.145","Text":"I have a plus in the exponent that tells me that to convert this to a multiplication."},{"Start":"03:10.145 ","End":"03:12.385","Text":"This is e^x,"},{"Start":"03:12.385 ","End":"03:15.320","Text":"e^1 by the formula,"},{"Start":"03:15.320 ","End":"03:20.825","Text":"then minus e^x plus e=0."},{"Start":"03:20.825 ","End":"03:22.805","Text":"Once again, a substitution."},{"Start":"03:22.805 ","End":"03:27.800","Text":"The same substitution, y equals e^ x."},{"Start":"03:27.800 ","End":"03:31.850","Text":"That will give me here y squared minus,"},{"Start":"03:31.850 ","End":"03:33.455","Text":"I\u0027ll put the e first,"},{"Start":"03:33.455 ","End":"03:40.985","Text":"e^1 is e times y minus y plus e=0."},{"Start":"03:40.985 ","End":"03:45.365","Text":"I just intended to combine these already. Let\u0027s do it now."},{"Start":"03:45.365 ","End":"03:48.890","Text":"Y^2 minus, how many y\u0027s do I have?"},{"Start":"03:48.890 ","End":"03:51.200","Text":"Minus e and minus 1."},{"Start":"03:51.200 ","End":"04:00.435","Text":"But I could take the minus out and say it\u0027s minus e plus 1 times y plus e=0."},{"Start":"04:00.435 ","End":"04:02.685","Text":"Instead of minus e minus 1,"},{"Start":"04:02.685 ","End":"04:04.660","Text":"I put minus of e plus 1."},{"Start":"04:04.660 ","End":"04:08.440","Text":"This is a quadratic equation in y."},{"Start":"04:08.440 ","End":"04:12.800","Text":"Let\u0027s solve it using the formula y equals minus b,"},{"Start":"04:12.800 ","End":"04:20.195","Text":"which is e plus 1 plus or minus the square root of b^2,"},{"Start":"04:20.195 ","End":"04:22.379","Text":"that\u0027s e plus 1^2."},{"Start":"04:22.379 ","End":"04:23.685","Text":"This is just a number,"},{"Start":"04:23.685 ","End":"04:26.975","Text":"e could\u0027ve been Phi or any other number."},{"Start":"04:26.975 ","End":"04:34.260","Text":"Minus 4ac, minus 4 times 1 times e/ 2a,"},{"Start":"04:34.260 ","End":"04:35.965","Text":"2 times 1 is 2."},{"Start":"04:35.965 ","End":"04:38.989","Text":"Let\u0027s see what\u0027s under the square root sign."},{"Start":"04:38.989 ","End":"04:40.340","Text":"I\u0027ll do this at the side."},{"Start":"04:40.340 ","End":"04:46.115","Text":"I have e plus 1^2 using the binomial expansion is"},{"Start":"04:46.115 ","End":"04:52.735","Text":"e^2 plus 2e plus 1 and then minus 4e."},{"Start":"04:52.735 ","End":"05:02.250","Text":"This equals e62 plus 2e minus 4e is minus 2e also plus 1."},{"Start":"05:02.250 ","End":"05:05.704","Text":"Again, I have a binomial product."},{"Start":"05:05.704 ","End":"05:08.855","Text":"This time it\u0027s e minus 1^2."},{"Start":"05:08.855 ","End":"05:11.705","Text":"I\u0027m not sure, just check it out by multiplying this out,"},{"Start":"05:11.705 ","End":"05:13.915","Text":"and this is what we get."},{"Start":"05:13.915 ","End":"05:20.180","Text":"Back to here, what we get is e plus 1 plus or minus,"},{"Start":"05:20.180 ","End":"05:23.480","Text":"and the square root of this thing is the square root of this thing."},{"Start":"05:23.480 ","End":"05:26.930","Text":"The square root of this is just e minus 1."},{"Start":"05:26.930 ","End":"05:28.730","Text":"Put it in brackets though."},{"Start":"05:28.730 ","End":"05:30.140","Text":"E minus 1 is positive."},{"Start":"05:30.140 ","End":"05:31.700","Text":"I said that e is bigger than 2,"},{"Start":"05:31.700 ","End":"05:33.170","Text":"it\u0027s certainly bigger than 1."},{"Start":"05:33.170 ","End":"05:37.640","Text":"Otherwise we have to take the positive turn over 2."},{"Start":"05:37.640 ","End":"05:39.140","Text":"Let\u0027s see what we get."},{"Start":"05:39.140 ","End":"05:40.864","Text":"If we take the plus,"},{"Start":"05:40.864 ","End":"05:44.550","Text":"we get e plus 1 plus e minus 1."},{"Start":"05:44.550 ","End":"05:47.940","Text":"I make that 2e/ 2,"},{"Start":"05:47.940 ","End":"05:51.770","Text":"and that is equal to e. If I take the minus,"},{"Start":"05:51.770 ","End":"05:53.600","Text":"I get e plus 1,"},{"Start":"05:53.600 ","End":"05:56.645","Text":"minus e plus 1."},{"Start":"05:56.645 ","End":"06:00.710","Text":"That gives me 2/2, which is 1."},{"Start":"06:00.710 ","End":"06:06.615","Text":"Now this is the solution for y. Y can be e or y can be 1."},{"Start":"06:06.615 ","End":"06:08.955","Text":"If I want x,"},{"Start":"06:08.955 ","End":"06:13.950","Text":"I can say that e^x equals e,"},{"Start":"06:13.950 ","End":"06:18.220","Text":"or e^x equals 1."},{"Start":"06:18.220 ","End":"06:21.860","Text":"Now it doesn\u0027t even matter what e is because whatever it is,"},{"Start":"06:21.860 ","End":"06:24.605","Text":"this is equal to e^1."},{"Start":"06:24.605 ","End":"06:26.165","Text":"In the other case,"},{"Start":"06:26.165 ","End":"06:31.945","Text":"I can write 1 as e^0 using this again."},{"Start":"06:31.945 ","End":"06:36.170","Text":"Each of these cases I have base e and base e,"},{"Start":"06:36.170 ","End":"06:37.865","Text":"so x equals 1."},{"Start":"06:37.865 ","End":"06:39.530","Text":"Base e and base e,"},{"Start":"06:39.530 ","End":"06:41.480","Text":"so x equals 0. Write it out."},{"Start":"06:41.480 ","End":"06:45.320","Text":"This one gives me x equals 1 by comparing exponents."},{"Start":"06:45.320 ","End":"06:48.200","Text":"This one gives me x equals 0."},{"Start":"06:48.200 ","End":"06:51.410","Text":"These are the 2 solutions for x,"},{"Start":"06:51.410 ","End":"06:53.240","Text":"and this is part b."},{"Start":"06:53.240 ","End":"06:56.130","Text":"We are done."}],"ID":8141}],"Thumbnail":null,"ID":56157},{"Name":"Scientific Notation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"intro","Duration":"10m 4s","ChapterTopicVideoID":8096,"CourseChapterTopicPlaylistID":56158,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8096.jpeg","UploadDate":"2020-09-30T13:56:37.0830000","DurationForVideoObject":"PT10M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"Scientific notation, which is a way of"},{"Start":"00:03.420 ","End":"00:08.685","Text":"writing very large or very small numbers or in fact any number,"},{"Start":"00:08.685 ","End":"00:13.755","Text":"but usually large or small and it\u0027s an alternative to decimal notation."},{"Start":"00:13.755 ","End":"00:15.330","Text":"In a moment I\u0027ll give an example,"},{"Start":"00:15.330 ","End":"00:16.800","Text":"I\u0027ll just define what it is."},{"Start":"00:16.800 ","End":"00:23.910","Text":"Scientific notation is when I write a number in the form m times 10 to"},{"Start":"00:23.910 ","End":"00:31.290","Text":"the power of n.There are names for m and n. M is called the 1 of 2 things."},{"Start":"00:31.290 ","End":"00:34.470","Text":"It\u0027s either called the mantissa,"},{"Start":"00:34.470 ","End":"00:38.445","Text":"it\u0027s also called the significant."},{"Start":"00:38.445 ","End":"00:42.210","Text":"M can be any real number,"},{"Start":"00:42.210 ","End":"00:49.920","Text":"positive or negative and n is called the exponent but it has to be an integer."},{"Start":"00:49.920 ","End":"00:54.825","Text":"Positive or negative, a whole number. I\u0027ll give an example."},{"Start":"00:54.825 ","End":"00:58.860","Text":"If we take the regular decimal number 350,"},{"Start":"00:58.860 ","End":"01:02.820","Text":"we could write it in scientific notation in more than 1 way."},{"Start":"01:02.820 ","End":"01:08.160","Text":"3.5 times 10^2."},{"Start":"01:08.160 ","End":"01:09.420","Text":"Because if you think about it,"},{"Start":"01:09.420 ","End":"01:12.870","Text":"10^2 squared is 100 and 3.5 times 100 is this."},{"Start":"01:12.870 ","End":"01:16.455","Text":"But I could also write it as 35 times 10,"},{"Start":"01:16.455 ","End":"01:18.925","Text":"but I\u0027ll write it as 10^1."},{"Start":"01:18.925 ","End":"01:24.135","Text":"In fact, I can also just write it as 350 times 1,"},{"Start":"01:24.135 ","End":"01:27.480","Text":"but that would just be 10^0."},{"Start":"01:27.480 ","End":"01:29.325","Text":"But in each of these cases,"},{"Start":"01:29.325 ","End":"01:35.010","Text":"this would be the mantissa or significant this part,"},{"Start":"01:35.010 ","End":"01:38.640","Text":"and the exponent would be this part here."},{"Start":"01:38.640 ","End":"01:40.500","Text":"I can only give 1 more example."},{"Start":"01:40.500 ","End":"01:41.730","Text":"It\u0027s a bit artificial,"},{"Start":"01:41.730 ","End":"01:42.945","Text":"but I want to make a point."},{"Start":"01:42.945 ","End":"01:49.260","Text":"I could also even write this as 3,500 divided by 10 or"},{"Start":"01:49.260 ","End":"01:56.955","Text":"times 1/10 or times 10 to the minus 1.The exponent can also be negative."},{"Start":"01:56.955 ","End":"02:01.080","Text":"In fact, the mantissa can also be negative."},{"Start":"02:01.080 ","End":"02:04.680","Text":"I mean, if I started off say with minus 350,"},{"Start":"02:04.680 ","End":"02:07.665","Text":"then I\u0027ve just put a minus everywhere."},{"Start":"02:07.665 ","End":"02:12.210","Text":"Mantissa is any number, the exponent,"},{"Start":"02:12.210 ","End":"02:16.550","Text":"an integer positive or negative and I\u0027m going to talk about something"},{"Start":"02:16.550 ","End":"02:21.365","Text":"called normalized scientific notation, we normalized form."},{"Start":"02:21.365 ","End":"02:31.410","Text":"This is what we are going to be using and that is when m is between 1 and 10,"},{"Start":"02:31.410 ","End":"02:35.415","Text":"actually strictly less than 10 but bigger or equal to 1."},{"Start":"02:35.415 ","End":"02:37.335","Text":"If you look at all these,"},{"Start":"02:37.335 ","End":"02:43.380","Text":"this is the 1 which is in normalized form because 3.5 is"},{"Start":"02:43.380 ","End":"02:46.890","Text":"between 1 and 10 so this is the"},{"Start":"02:46.890 ","End":"02:50.100","Text":"normalized and there\u0027s something I said which is not quite"},{"Start":"02:50.100 ","End":"02:53.595","Text":"accurate because if we took the case with the minuses,"},{"Start":"02:53.595 ","End":"02:57.885","Text":"then minus 3.5 isn\u0027t between 1 and 10."},{"Start":"02:57.885 ","End":"02:59.805","Text":"But if I put an absolute value,"},{"Start":"02:59.805 ","End":"03:01.380","Text":"meaning ignore the sign,"},{"Start":"03:01.380 ","End":"03:02.985","Text":"it may or may not be a minus."},{"Start":"03:02.985 ","End":"03:06.045","Text":"This bit has to be between 1 and 10,"},{"Start":"03:06.045 ","End":"03:07.740","Text":"whether we have the minus or not."},{"Start":"03:07.740 ","End":"03:12.660","Text":"This is a normalized scientific notation and in fact,"},{"Start":"03:12.660 ","End":"03:14.760","Text":"we\u0027re not going to be considering any other kind."},{"Start":"03:14.760 ","End":"03:16.845","Text":"If I just say scientific notation,"},{"Start":"03:16.845 ","End":"03:21.105","Text":"I\u0027ll usually or in this course will mean it to be normalized."},{"Start":"03:21.105 ","End":"03:25.200","Text":"In the United Kingdom it\u0027s not called normalized."},{"Start":"03:25.200 ","End":"03:29.160","Text":"It\u0027s usually called in the standard form."},{"Start":"03:29.160 ","End":"03:34.665","Text":"Some examples now of how to convert from decimal to scientific and the other way round."},{"Start":"03:34.665 ","End":"03:39.015","Text":"I brought a few examples that I found on the Internet and Wikipedia,"},{"Start":"03:39.015 ","End":"03:42.090","Text":"and I\u0027ll show you how we do the conversions."},{"Start":"03:42.090 ","End":"03:45.990","Text":"Let\u0027s first of all go from decimal to scientific."},{"Start":"03:45.990 ","End":"03:49.200","Text":"Decimal to scientific,"},{"Start":"03:49.200 ","End":"03:52.065","Text":"and just take this as an example."},{"Start":"03:52.065 ","End":"03:53.550","Text":"Let\u0027s leave out the minus."},{"Start":"03:53.550 ","End":"03:56.940","Text":"I\u0027ll take 350 and we make sure"},{"Start":"03:56.940 ","End":"04:00.660","Text":"to include the decimal point I could write point 0 doesn\u0027t matter."},{"Start":"04:00.660 ","End":"04:02.820","Text":"The decimal point is here."},{"Start":"04:02.820 ","End":"04:04.950","Text":"Then we move the decimal point."},{"Start":"04:04.950 ","End":"04:08.820","Text":"In this case it will be 2 to the left in such a way"},{"Start":"04:08.820 ","End":"04:12.990","Text":"that what you get is something between 1 and 10,"},{"Start":"04:12.990 ","End":"04:14.940","Text":"3.5 in this case,"},{"Start":"04:14.940 ","End":"04:16.390","Text":"and I\u0027m counting 1,"},{"Start":"04:16.390 ","End":"04:18.810","Text":"2 to the left."},{"Start":"04:18.810 ","End":"04:22.965","Text":"Then this becomes 3.5 is"},{"Start":"04:22.965 ","End":"04:29.295","Text":"the mantissa or significant and then 10 to the power of and when we move to the left,"},{"Start":"04:29.295 ","End":"04:32.690","Text":"we put a positive 2 here and now I\u0027ll give"},{"Start":"04:32.690 ","End":"04:36.835","Text":"an example where we move to the right and see if I can find a good 1 here."},{"Start":"04:36.835 ","End":"04:38.700","Text":"Notice we have 1 that\u0027s very similar,"},{"Start":"04:38.700 ","End":"04:41.775","Text":"like 300 decimal point would be here."},{"Start":"04:41.775 ","End":"04:43.305","Text":"Move it 1, 2,"},{"Start":"04:43.305 ","End":"04:46.380","Text":"and we\u0027ve got 3-point note times 10^2,"},{"Start":"04:46.380 ","End":"04:48.375","Text":"where we just write it as 3."},{"Start":"04:48.375 ","End":"04:50.835","Text":"The other example would be,"},{"Start":"04:50.835 ","End":"04:52.800","Text":"this will be a good example."},{"Start":"04:52.800 ","End":"05:00.820","Text":"Write it as 4,321,768."},{"Start":"05:00.820 ","End":"05:03.860","Text":"What I would do would be,"},{"Start":"05:03.860 ","End":"05:06.590","Text":"I was going to give an example of the other way. Never mind."},{"Start":"05:06.590 ","End":"05:11.570","Text":"It\u0027s to the left also 1,"},{"Start":"05:11.570 ","End":"05:12.655","Text":"2, 3, 4, 5, 6."},{"Start":"05:12.655 ","End":"05:14.632","Text":"You can even number them 1, 2, 3, 4, 5,"},{"Start":"05:14.632 ","End":"05:21.360","Text":"6 and then the decimal place is now here and so"},{"Start":"05:21.360 ","End":"05:30.045","Text":"it\u0027s 4.321768 times 10^6."},{"Start":"05:30.045 ","End":"05:34.440","Text":"Now I\u0027ll give an example of the negative exponent here, this last 1."},{"Start":"05:34.440 ","End":"05:37.510","Text":"Suppose I took 0.000000751."},{"Start":"05:45.320 ","End":"05:49.770","Text":"What we do to get it to a number between 1 and 10 if we"},{"Start":"05:49.770 ","End":"05:53.589","Text":"want it to be 7.51 so we move this 1,"},{"Start":"05:53.589 ","End":"05:55.506","Text":"2, 3, 4,"},{"Start":"05:55.506 ","End":"05:59.325","Text":"5, 6, 7, 8, 9."},{"Start":"05:59.325 ","End":"06:03.750","Text":"I\u0027m not going to number them all but there\u0027s 9 of them."},{"Start":"06:03.750 ","End":"06:08.040","Text":"1, 2 up to 9 and so we take the"},{"Start":"06:08.040 ","End":"06:14.010","Text":"7.51 and 10 to the power of them because we move to the right,"},{"Start":"06:14.010 ","End":"06:15.315","Text":"not to the left,"},{"Start":"06:15.315 ","End":"06:17.625","Text":"then it\u0027s minus 9."},{"Start":"06:17.625 ","End":"06:19.050","Text":"In these cases,"},{"Start":"06:19.050 ","End":"06:21.615","Text":"we move to the left, so it\u0027s positive exponent."},{"Start":"06:21.615 ","End":"06:23.760","Text":"When we had to move to the right,"},{"Start":"06:23.760 ","End":"06:26.325","Text":"it\u0027s a negative exponent and that\u0027s written here."},{"Start":"06:26.325 ","End":"06:33.975","Text":"Now let\u0027s go in the other direction from scientific to decimal."},{"Start":"06:33.975 ","End":"06:36.180","Text":"Very similar to this, but backwards."},{"Start":"06:36.180 ","End":"06:38.340","Text":"As an example, I\u0027ll take this 1."},{"Start":"06:38.340 ","End":"06:45.225","Text":"Suppose I had minus 5.3 times 10^4."},{"Start":"06:45.225 ","End":"06:47.985","Text":"Well, the sign, if there\u0027s any sign just stays,"},{"Start":"06:47.985 ","End":"06:52.980","Text":"and then I have a 5 and a 3."},{"Start":"06:52.980 ","End":"06:55.230","Text":"The decimal point starts here,"},{"Start":"06:55.230 ","End":"07:00.360","Text":"and then we move it because 4 is a positive number to the right,"},{"Start":"07:00.360 ","End":"07:02.562","Text":"1, 2, 3,"},{"Start":"07:02.562 ","End":"07:04.755","Text":"4, 1, 2, 3, 4."},{"Start":"07:04.755 ","End":"07:08.460","Text":"Then we have to fill in the blanks with the 0s."},{"Start":"07:08.460 ","End":"07:11.280","Text":"Put a 0 here, 0, 0."},{"Start":"07:11.280 ","End":"07:21.495","Text":"Now that gives us minus 5,3000 and I don\u0027t need to put the decimal point there."},{"Start":"07:21.495 ","End":"07:24.450","Text":"Another example, let\u0027s take this 1,"},{"Start":"07:24.450 ","End":"07:27.795","Text":"2 times 10 to the minus 1,"},{"Start":"07:27.795 ","End":"07:31.020","Text":"so 2 times 10 to the minus 1."},{"Start":"07:31.020 ","End":"07:33.870","Text":"What I do is there\u0027s a decimal point here,"},{"Start":"07:33.870 ","End":"07:38.880","Text":"it\u0027s 2. and then because of the minus 1,"},{"Start":"07:38.880 ","End":"07:42.960","Text":"I move it 1 to the left this time."},{"Start":"07:42.960 ","End":"07:48.675","Text":"So decimal point is now here so the answer is 0.2."},{"Start":"07:48.675 ","End":"07:50.145","Text":"You could leave it like that,"},{"Start":"07:50.145 ","End":"07:53.925","Text":"but it\u0027s customary to put a 0 in front of it."},{"Start":"07:53.925 ","End":"07:56.355","Text":"Maybe I\u0027ll do 1 more that\u0027s not here."},{"Start":"07:56.355 ","End":"07:59.040","Text":"Let\u0027s take another negative example."},{"Start":"07:59.040 ","End":"08:03.100","Text":"I don\u0027t know, 12.34,"},{"Start":"08:03.110 ","End":"08:06.930","Text":"say times 10 to the,"},{"Start":"08:06.930 ","End":"08:08.400","Text":"I will take minus 1 again,"},{"Start":"08:08.400 ","End":"08:12.405","Text":"minus 6 and let\u0027s see what this gives us."},{"Start":"08:12.405 ","End":"08:18.975","Text":"We start off with the minus and then we have 12.34."},{"Start":"08:18.975 ","End":"08:23.940","Text":"But we move the decimal 0.6 to the left,"},{"Start":"08:23.940 ","End":"08:30.630","Text":"so 1,2,3,4,5,6 and the minus"},{"Start":"08:30.630 ","End":"08:36.375","Text":"is misplaced and then you put zeros in the missing places."},{"Start":"08:36.375 ","End":"08:41.265","Text":"Decimal point is now here and so this is equal to minus."},{"Start":"08:41.265 ","End":"08:43.500","Text":"Again, we don\u0027t just write the decimal point,"},{"Start":"08:43.500 ","End":"08:49.300","Text":"it\u0027s 0 point and then 00001234."},{"Start":"08:53.870 ","End":"08:57.315","Text":"That\u0027s enough examples for conversion."},{"Start":"08:57.315 ","End":"08:59.760","Text":"I wanted to say something about calculators."},{"Start":"08:59.760 ","End":"09:04.320","Text":"There is scientific notation on most scientific calculators,"},{"Start":"09:04.320 ","End":"09:07.260","Text":"but each 1 is so different in which buttons you press."},{"Start":"09:07.260 ","End":"09:09.060","Text":"I\u0027m not going to get into it deeply."},{"Start":"09:09.060 ","End":"09:11.415","Text":"I\u0027ll just say something about the display,"},{"Start":"09:11.415 ","End":"09:14.505","Text":"the way these numbers are displayed."},{"Start":"09:14.505 ","End":"09:16.620","Text":"What it will do, for example,"},{"Start":"09:16.620 ","End":"09:25.635","Text":"suppose I take the number 6.02 times 10^23."},{"Start":"09:25.635 ","End":"09:29.750","Text":"If you\u0027ve studied chemistry now this is Avogadro\u0027s constant."},{"Start":"09:29.750 ","End":"09:35.630","Text":"How it will appear on the calculator is usually in every calculator is different,"},{"Start":"09:35.630 ","End":"09:41.570","Text":"but usually it doesn\u0027t let her E between the mantissa"},{"Start":"09:41.570 ","End":"09:48.095","Text":"and the exponent and this E means times 10 to the power of."},{"Start":"09:48.095 ","End":"09:52.685","Text":"That\u0027s just because it\u0027s convenient for the calculator display to do it this way."},{"Start":"09:52.685 ","End":"09:59.150","Text":"There are a lot of things you can do with scientific notation and computations,"},{"Start":"09:59.150 ","End":"10:05.390","Text":"but I\u0027ll leave that for the solved examples following this tutorial and I\u0027m done here."}],"ID":8237},{"Watched":false,"Name":"Exercise1","Duration":"4m 26s","ChapterTopicVideoID":8097,"CourseChapterTopicPlaylistID":56158,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8097.jpeg","UploadDate":"2020-09-30T13:59:06.6330000","DurationForVideoObject":"PT4M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"In this exercise, we have to multiply this times this over this."},{"Start":"00:04.065 ","End":"00:08.490","Text":"But we want to convert the numbers into scientific notation."},{"Start":"00:08.490 ","End":"00:12.390","Text":"Then at the end, express the answer in scientific notation."},{"Start":"00:12.390 ","End":"00:16.080","Text":"Just a quick reminder the scientific notation means writing"},{"Start":"00:16.080 ","End":"00:21.720","Text":"a number in the form m times 10^n,"},{"Start":"00:21.720 ","End":"00:24.300","Text":"where the m is called the Mantissa,"},{"Start":"00:24.300 ","End":"00:27.360","Text":"or the significant and n is the exponent"},{"Start":"00:27.360 ","End":"00:31.455","Text":"but we use what is called standardize scientific notation."},{"Start":"00:31.455 ","End":"00:36.010","Text":"Sorry, standardize is what it\u0027s called in UK, it\u0027s normalized,"},{"Start":"00:36.010 ","End":"00:41.280","Text":"in the US, which means that M is between 1 and 10."},{"Start":"00:41.280 ","End":"00:43.550","Text":"Let\u0027s convert each of these."},{"Start":"00:43.550 ","End":"00:47.540","Text":"I\u0027ll start with the 24 million and I won\u0027t bother writing"},{"Start":"00:47.540 ","End":"00:52.400","Text":"the commas and notice that the decimal point is here."},{"Start":"00:52.400 ","End":"00:54.590","Text":"Convert the scientific, we need to move"},{"Start":"00:54.590 ","End":"00:57.570","Text":"the decimal point to get a number between 1 and 10,"},{"Start":"00:57.570 ","End":"00:59.770","Text":"so we move it to the left,"},{"Start":"01:00.920 ","End":"01:03.690","Text":"1, 2, 3, 4, 5, 6, 7."},{"Start":"01:03.690 ","End":"01:06.550","Text":"Let\u0027s make a note to that 7 times,"},{"Start":"01:06.550 ","End":"01:09.590","Text":"I moved it and then it becomes 2.4."},{"Start":"01:09.590 ","End":"01:16.710","Text":"2.4 times, and because we moved it to the left, it\u0027s 10^7."},{"Start":"01:16.840 ","End":"01:19.235","Text":"Next, let\u0027s take the"},{"Start":"01:19.235 ","End":"01:28.165","Text":"0.009 and we want to move the decimal point also to make it between 1 and 10,"},{"Start":"01:28.165 ","End":"01:31.202","Text":"so we go 1, 2,"},{"Start":"01:31.202 ","End":"01:33.840","Text":"3, and now it\u0027s 9 point,"},{"Start":"01:33.840 ","End":"01:35.190","Text":"so it\u0027s between 1 and 10."},{"Start":"01:35.190 ","End":"01:43.065","Text":"We can write this as 9 times 10,"},{"Start":"01:43.065 ","End":"01:44.330","Text":"how many times did I move it?"},{"Start":"01:44.330 ","End":"01:45.918","Text":"It was 1,"},{"Start":"01:45.918 ","End":"01:47.655","Text":"2, 3, I should have written 3."},{"Start":"01:47.655 ","End":"01:51.330","Text":"10^minus 3 because I moved to the right."},{"Start":"01:51.330 ","End":"01:59.904","Text":"The last one, 0.00015 is like the one above just 1,"},{"Start":"01:59.904 ","End":"02:01.960","Text":"2, 3, 4."},{"Start":"02:01.960 ","End":"02:04.305","Text":"I move it 4 places,"},{"Start":"02:04.305 ","End":"02:11.410","Text":"and that gives me 1.5 times 10^minus 4."},{"Start":"02:11.410 ","End":"02:13.880","Text":"Now I\u0027m going to substitute them here."},{"Start":"02:13.880 ","End":"02:21.190","Text":"This becomes, I will continue here, 2.4 times 10^7."},{"Start":"02:21.190 ","End":"02:25.520","Text":"I\u0027ll use cross multiplication because I don\u0027t want to confuse it with the decimal point."},{"Start":"02:25.520 ","End":"02:28.249","Text":"Next one is 9 times"},{"Start":"02:28.249 ","End":"02:36.365","Text":"10^minus 3 over 1.5 times 10^minus 4."},{"Start":"02:36.365 ","End":"02:40.025","Text":"I can split this up into 2 separate products,"},{"Start":"02:40.025 ","End":"02:43.010","Text":"the Mantissa part separately,"},{"Start":"02:43.010 ","End":"02:45.934","Text":"the 10 to the power of the exponent separately,"},{"Start":"02:45.934 ","End":"02:53.565","Text":"so we get 2.4 times 9 over 1.5"},{"Start":"02:53.565 ","End":"03:03.190","Text":"times 10^7 times 10^minus 3 over 10^minus 4."},{"Start":"03:03.190 ","End":"03:06.440","Text":"Now this part we can do on a regular calculator,"},{"Start":"03:06.440 ","End":"03:11.205","Text":"and I make it 14.4 and then here,"},{"Start":"03:11.205 ","End":"03:17.285","Text":"we use the rules of exponents to figure out what this is in terms of 10 to the power of."},{"Start":"03:17.285 ","End":"03:19.610","Text":"What we do is for multiplication,"},{"Start":"03:19.610 ","End":"03:22.550","Text":"we do use addition and for division and subtraction,"},{"Start":"03:22.550 ","End":"03:29.280","Text":"so we get 7 plus negative 3 minus negative 4."},{"Start":"03:29.280 ","End":"03:31.530","Text":"So this comes out,"},{"Start":"03:31.530 ","End":"03:34.515","Text":"7 minus 3 is 4,"},{"Start":"03:34.515 ","End":"03:36.870","Text":"4 minus minus 4 is 8."},{"Start":"03:36.870 ","End":"03:43.230","Text":"I would get 14.4 times 10^8."},{"Start":"03:43.230 ","End":"03:46.600","Text":"But this is not good enough,"},{"Start":"03:46.600 ","End":"03:53.560","Text":"because we\u0027re working in normalized scientific notation and this is not between 1 and 10."},{"Start":"03:53.560 ","End":"03:57.500","Text":"What we do is we make a fix and write this in"},{"Start":"03:57.500 ","End":"04:01.580","Text":"scientific notation again as move decimal point here,"},{"Start":"04:01.580 ","End":"04:06.575","Text":"so we get 1.44 times 10^1,"},{"Start":"04:06.575 ","End":"04:12.770","Text":"and then times 10^8 and then we can combine these and finally"},{"Start":"04:12.770 ","End":"04:20.155","Text":"get it as 1.44 times 10^1 plus 8 is 9."},{"Start":"04:20.155 ","End":"04:27.000","Text":"This is the answer in normalized scientific notation. We\u0027re done."}],"ID":8238},{"Watched":false,"Name":"Exercise2","Duration":"3m 4s","ChapterTopicVideoID":8098,"CourseChapterTopicPlaylistID":56158,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8098.jpeg","UploadDate":"2020-09-30T14:00:48.2000000","DurationForVideoObject":"PT3M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"Here we have exercise in astronomy,"},{"Start":"00:03.600 ","End":"00:07.500","Text":"but it\u0027s really just a matter of scientific notation."},{"Start":"00:07.500 ","End":"00:09.195","Text":"We\u0027re given the speed of light,"},{"Start":"00:09.195 ","End":"00:12.180","Text":"it\u0027s 1.86 times 10^5,"},{"Start":"00:12.180 ","End":"00:14.670","Text":"and Mars is,"},{"Start":"00:14.670 ","End":"00:18.330","Text":"in scientific notation, this is the distance from the sun."},{"Start":"00:18.330 ","End":"00:23.339","Text":"The question is, how long does it take light from the sun to reach Mars?"},{"Start":"00:23.339 ","End":"00:25.740","Text":"We want the answer in scientific notation."},{"Start":"00:25.740 ","End":"00:31.025","Text":"We know that distance is speed times time."},{"Start":"00:31.025 ","End":"00:32.660","Text":"So if I want time,"},{"Start":"00:32.660 ","End":"00:34.640","Text":"it\u0027s distance over speed."},{"Start":"00:34.640 ","End":"00:44.525","Text":"Time is distance over speed and so time t is distance,"},{"Start":"00:44.525 ","End":"00:52.325","Text":"which is given here 1.416 times 10^8"},{"Start":"00:52.325 ","End":"01:01.520","Text":"divided by the speed which is 1.86 times 10^5."},{"Start":"01:01.520 ","End":"01:03.200","Text":"Now, what we do is we separate it."},{"Start":"01:03.200 ","End":"01:07.220","Text":"We can say that it\u0027s 1.416 over"},{"Start":"01:07.220 ","End":"01:15.145","Text":"1.86 times 10^8 over 10^5."},{"Start":"01:15.145 ","End":"01:19.270","Text":"This comes out to 0.76."},{"Start":"01:19.270 ","End":"01:26.230","Text":"I took it just to 4 decimal places and this part comes out to be 10^3."},{"Start":"01:26.230 ","End":"01:30.535","Text":"I got the 3 by doing 8 minus 5 equals 3."},{"Start":"01:30.535 ","End":"01:37.390","Text":"Now this is not normalized scientific notation because this is not between 1 and 10 so I"},{"Start":"01:37.390 ","End":"01:44.305","Text":"rewrite it as 7.613 because I move the decimal point one to the right,"},{"Start":"01:44.305 ","End":"01:48.175","Text":"I have to multiply by 10 to the minus 1."},{"Start":"01:48.175 ","End":"01:54.690","Text":"That\u0027s this bit, and then there\u0027s still the 10^3 and so the answer is"},{"Start":"01:54.690 ","End":"02:03.130","Text":"7.613 times 10^2 and"},{"Start":"02:03.130 ","End":"02:07.655","Text":"because we\u0027re asked for unit of time and we\u0027re working in seconds,"},{"Start":"02:07.655 ","End":"02:14.850","Text":"we should write seconds and I\u0027ll highlight the answer and we\u0027re done."},{"Start":"02:14.920 ","End":"02:18.710","Text":"I\u0027m just curious what this is in minutes and seconds."},{"Start":"02:18.710 ","End":"02:21.680","Text":"I could say if I didn\u0027t want to use scientific notation,"},{"Start":"02:21.680 ","End":"02:23.585","Text":"if I multiply this by 100,"},{"Start":"02:23.585 ","End":"02:29.690","Text":"I would get 761.3 seconds."},{"Start":"02:29.690 ","End":"02:33.425","Text":"Let\u0027s just say we going to round it off to whole numbers,"},{"Start":"02:33.425 ","End":"02:38.630","Text":"so let\u0027s take 761 and I want to know what that is in minutes and seconds."},{"Start":"02:38.630 ","End":"02:41.000","Text":"If I divide it by 60,"},{"Start":"02:41.000 ","End":"02:46.685","Text":"then I get basically 12 with remainder 41."},{"Start":"02:46.685 ","End":"02:53.390","Text":"Let\u0027s see, 12 times 60 is 720 and remainder 41 or 41.3."},{"Start":"02:53.390 ","End":"02:58.010","Text":"So it\u0027s approximately 12 minutes and"},{"Start":"02:58.010 ","End":"03:05.399","Text":"41 seconds to take light to reach from the sun to Mars."}],"ID":8239},{"Watched":false,"Name":"Exercise3","Duration":"3m 45s","ChapterTopicVideoID":8099,"CourseChapterTopicPlaylistID":56158,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8099.jpeg","UploadDate":"2020-09-30T14:02:48.1370000","DurationForVideoObject":"PT3M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.160","Text":"Here we have an exercise in scientific notation but applied to astronomy."},{"Start":"00:05.160 ","End":"00:09.975","Text":"We\u0027re talking about the mass of Pluto used to be a planet,"},{"Start":"00:09.975 ","End":"00:12.210","Text":"now I think it\u0027s called a dwarf planet or something."},{"Start":"00:12.210 ","End":"00:16.845","Text":"Anyway, we\u0027re given its mass in kilograms as this,"},{"Start":"00:16.845 ","End":"00:18.849","Text":"and we\u0027re also given the exchange,"},{"Start":"00:18.849 ","End":"00:22.496","Text":"the conversion rate between tons and kilograms,"},{"Start":"00:22.496 ","End":"00:26.040","Text":"that 1 ton is this many kilograms."},{"Start":"00:26.040 ","End":"00:30.899","Text":"We want to compute the mass of Pluto in tons."},{"Start":"00:30.899 ","End":"00:34.140","Text":"We want the answer in scientific notation, strictly speaking,"},{"Start":"00:34.140 ","End":"00:38.190","Text":"which leads to use the word weight in both of the molar mass in both of them are,"},{"Start":"00:38.190 ","End":"00:40.500","Text":"though mass is more correct,"},{"Start":"00:40.500 ","End":"00:42.205","Text":"so let\u0027s be precise."},{"Start":"00:42.205 ","End":"00:47.420","Text":"Now, clearly, what we want to do is take this number and the question is,"},{"Start":"00:47.420 ","End":"00:52.685","Text":"do we multiply or divide by this 888.9?"},{"Start":"00:52.685 ","End":"00:55.610","Text":"There\u0027 s an easy way to remember this."},{"Start":"00:55.610 ","End":"01:02.600","Text":"Well, let me write 1.3 times 10^22 and kilograms is abbreviated like this."},{"Start":"01:02.600 ","End":"01:06.590","Text":"We\u0027re going to multiply by something which is equal to 1."},{"Start":"01:06.590 ","End":"01:10.475","Text":"Now, since I want the kilograms to disappear and the terms to appear,"},{"Start":"01:10.475 ","End":"01:13.058","Text":"I\u0027ll put on the numerator term,"},{"Start":"01:13.058 ","End":"01:15.915","Text":"and in the denominator kilograms."},{"Start":"01:15.915 ","End":"01:24.210","Text":"I\u0027m going to put 1 ton is equal to 888.9 kilograms,"},{"Start":"01:24.210 ","End":"01:27.880","Text":"then the kilograms cancel and you get your answer in tons."},{"Start":"01:27.880 ","End":"01:30.880","Text":"That\u0027s how you can remember how in many scientific problems,"},{"Start":"01:30.880 ","End":"01:32.949","Text":"whether you multiply or divide."},{"Start":"01:32.949 ","End":"01:39.130","Text":"One other thing, the word ton is different in the United States and the United Kingdom,"},{"Start":"01:39.130 ","End":"01:43.345","Text":"this is true for the United States."},{"Start":"01:43.345 ","End":"01:47.590","Text":"In the UK, a ton is bigger than an American ton."},{"Start":"01:47.590 ","End":"01:50.365","Text":"It wouldn\u0027t apply, so these are US tons."},{"Start":"01:50.365 ","End":"01:52.540","Text":"Now, we\u0027re going to divide this over this."},{"Start":"01:52.540 ","End":"01:54.790","Text":"This already is in scientific notation."},{"Start":"01:54.790 ","End":"02:01.705","Text":"Let\u0027s convert 888.9 to normalize scientific notation."},{"Start":"02:01.705 ","End":"02:03.080","Text":"What we do,"},{"Start":"02:03.080 ","End":"02:06.850","Text":"is shift the decimal point until we get a number between 1-10,"},{"Start":"02:06.850 ","End":"02:09.100","Text":"so e have to do is shift it 1,"},{"Start":"02:09.100 ","End":"02:11.120","Text":"2 to the left,"},{"Start":"02:11.120 ","End":"02:19.490","Text":"and then we can write it as 8.889 times 10^2,"},{"Start":"02:19.490 ","End":"02:23.405","Text":"because I move 2 to the left, it\u0027s 10^2."},{"Start":"02:23.405 ","End":"02:30.739","Text":"What we want is 1.3 times 10^22 divided"},{"Start":"02:30.739 ","End":"02:39.585","Text":"by 8.889 times 10 squared."},{"Start":"02:39.585 ","End":"02:41.700","Text":"This we can split up as"},{"Start":"02:41.700 ","End":"02:44.230","Text":"1.3/8.889"},{"Start":"02:45.800 ","End":"02:52.120","Text":"and then 10^22/10^2."},{"Start":"02:52.120 ","End":"02:54.845","Text":"This we can do on the calculator."},{"Start":"02:54.845 ","End":"03:00.320","Text":"It comes out. I\u0027ll just do it to 3 decimal places, 0.146."},{"Start":"03:00.320 ","End":"03:03.515","Text":"My calculator gave me and this over this, well,"},{"Start":"03:03.515 ","End":"03:06.740","Text":"22 minus 2 is 20,"},{"Start":"03:06.740 ","End":"03:09.400","Text":"so it\u0027s times 10^20."},{"Start":"03:09.400 ","End":"03:13.220","Text":"Now, this is not normalized scientific notation."},{"Start":"03:13.220 ","End":"03:16.430","Text":"We want is something here between 1 and 10."},{"Start":"03:16.430 ","End":"03:19.265","Text":"If I bring the decimal point to the right,"},{"Start":"03:19.265 ","End":"03:27.060","Text":"I can write this as 1.46 times 10^-1."},{"Start":"03:27.060 ","End":"03:30.010","Text":"Then times 10^20."},{"Start":"03:30.010 ","End":"03:32.000","Text":"I combine these 2,"},{"Start":"03:32.000 ","End":"03:38.565","Text":"I get 1.46 times 10^19,"},{"Start":"03:38.565 ","End":"03:41.400","Text":"and the answer is in American tons."},{"Start":"03:41.400 ","End":"03:46.000","Text":"Just highlight it and we are done."}],"ID":8240}],"Thumbnail":null,"ID":56158}]

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