1 00:00:00,000 --> 00:00:00,500 2 00:00:00,500 --> 00:00:02,480 Let's do some exponent examples 3 00:00:02,480 --> 00:00:04,419 that involve division. 4 00:00:04,419 --> 00:00:08,339 Let's say I were to ask you what 5 to the sixth power 5 00:00:08,339 --> 00:00:12,370 divided by 5 to the second power is? 6 00:00:12,369 --> 00:00:15,699 Well, we can just go to the basic definition of what an 7 00:00:15,699 --> 00:00:19,419 exponent represents and say 5 to the sixth power, that's 8 00:00:19,420 --> 00:00:26,470 going to be 5 times 5 times 5 times 5 times 5-- one 9 00:00:26,469 --> 00:00:27,730 more 5-- times 5. 10 00:00:27,730 --> 00:00:30,030 5 times itself six times. 11 00:00:30,030 --> 00:00:33,719 And 5 squared, that's just 5 times itself two times, so 12 00:00:33,719 --> 00:00:36,769 it's just going to be 5 times 5. 13 00:00:36,770 --> 00:00:40,380 Well, we know how to simplify a fraction or a rational 14 00:00:40,380 --> 00:00:41,620 expression like this. 15 00:00:41,619 --> 00:00:45,469 We can divide the numerator and the denominator by one 5, 16 00:00:45,469 --> 00:00:47,899 and then these will cancel out, and then we can do it by 17 00:00:47,899 --> 00:00:51,109 another 5, or this 5 and this 5 will cancel out. 18 00:00:51,109 --> 00:00:52,960 And what are we going to be left with? 19 00:00:52,960 --> 00:00:58,219 5 times 5 times 5 times 5 over 1, or you could say that this 20 00:00:58,219 --> 00:01:01,539 is just 5 to the fourth power. 21 00:01:01,539 --> 00:01:03,320 Now, notice what happens. 22 00:01:03,320 --> 00:01:07,060 Essentially we started with six in the numerator, six 5's 23 00:01:07,060 --> 00:01:10,579 multiplied by themselves in the numerator, and then we 24 00:01:10,579 --> 00:01:11,719 subtracted out. 25 00:01:11,719 --> 00:01:14,519 We were able to cancel out the 2 in the denominator. 26 00:01:14,519 --> 00:01:21,449 So this really was equal to 5 to the sixth power minus 2. 27 00:01:21,450 --> 00:01:24,019 So we were able to subtract the exponent in the 28 00:01:24,019 --> 00:01:26,619 denominator from the exponent in the numerator. 29 00:01:26,620 --> 00:01:29,609 Let's remember how this relates to multiplication. 30 00:01:29,609 --> 00:01:32,790 If I had 5 to the-- let me do this in a different color. 31 00:01:32,790 --> 00:01:41,570 5 to the sixth times 5 to the second power, we saw in the 32 00:01:41,569 --> 00:01:48,329 last video that this is equal to 5 to the 6 plus-- I'm 33 00:01:48,329 --> 00:01:51,969 trying to make it color coded for you-- 6 plus 2 power. 34 00:01:51,969 --> 00:01:53,670 Now, we see a new property. 35 00:01:53,670 --> 00:01:55,370 And in the next video, we're going see that these aren't 36 00:01:55,370 --> 00:01:56,939 really different properties. 37 00:01:56,939 --> 00:01:59,209 They're really kind of same sides of the same coin when we 38 00:01:59,209 --> 00:02:01,519 learn about negative exponents. 39 00:02:01,519 --> 00:02:09,250 But now in this video, we just saw that 5 to the sixth power 40 00:02:09,250 --> 00:02:17,039 divided by 5 to the second power-- let me do it in a 41 00:02:17,039 --> 00:02:26,609 different color-- is going to be equal to 5 to the-- it's 42 00:02:26,610 --> 00:02:29,830 time consuming to make it color coded for you-- 6 minus 43 00:02:29,830 --> 00:02:35,580 2 power or 5 to the fourth power. 44 00:02:35,580 --> 00:02:37,420 Here it's going to be 5 to the eighth. 45 00:02:37,419 --> 00:02:40,094 So when you multiply exponents with the same base, you add 46 00:02:40,094 --> 00:02:40,830 the exponents. 47 00:02:40,830 --> 00:02:45,270 When you divide with the same base, you subtract the 48 00:02:45,270 --> 00:02:48,870 denominator exponent from the numerator exponent. 49 00:02:48,870 --> 00:02:54,250 Let's do a bunch more of these examples right here. 50 00:02:54,250 --> 00:02:59,469 What is 6 to the seventh power divided by 6 51 00:02:59,469 --> 00:03:01,710 to the third power? 52 00:03:01,710 --> 00:03:04,060 Well, once again, we can just use this property. 53 00:03:04,060 --> 00:03:08,189 This going to be 6 to the 7 minus 3 power, which is equal 54 00:03:08,189 --> 00:03:11,430 to 6 to the fourth power. 55 00:03:11,430 --> 00:03:13,540 And you can multiply it out this way like we did in the 56 00:03:13,539 --> 00:03:17,679 first problem and verify that it indeed will be 6 to the 57 00:03:17,680 --> 00:03:18,840 fourth power. 58 00:03:18,840 --> 00:03:22,460 Now let's try something interesting. 59 00:03:22,460 --> 00:03:26,349 This will be a good segue into the next video. 60 00:03:26,349 --> 00:03:32,139 Let's say we have 3 to the fourth power divided by 3 to 61 00:03:32,139 --> 00:03:33,959 the tenth power. 62 00:03:33,960 --> 00:03:37,240 Well, if we just go from basic principles, this would be 3 63 00:03:37,240 --> 00:03:43,100 times 3 times 3 times 3, all of that over 3 times 3-- we're 64 00:03:43,099 --> 00:03:47,039 going to have ten of these-- 3 times 3 times 3 times 3 65 00:03:47,039 --> 00:03:48,919 times 3 times 3. 66 00:03:48,919 --> 00:03:49,309 How many is that? 67 00:03:49,310 --> 00:03:54,960 One, two, three, four, five, six, seven, eight, nine, ten. 68 00:03:54,960 --> 00:03:58,159 Well, if we do what we did in the last video, this 3 cancels 69 00:03:58,159 --> 00:03:58,909 with that 3. 70 00:03:58,909 --> 00:03:59,789 Those 3's cancel. 71 00:03:59,789 --> 00:04:00,569 Those 3's cancel. 72 00:04:00,569 --> 00:04:01,739 Those 3's cancel. 73 00:04:01,740 --> 00:04:05,590 And we're left with 1 over-- one, two, three, 74 00:04:05,590 --> 00:04:07,729 four, five, six 3's. 75 00:04:07,729 --> 00:04:12,090 So 1 over 3 to the sixth power, right? 76 00:04:12,090 --> 00:04:14,969 We have 1 over all of these 3's down here. 77 00:04:14,969 --> 00:04:18,028 But that property that I just told you, would have told you 78 00:04:18,028 --> 00:04:24,819 that this should also be equal to 3 to the 4 minus 10 power. 79 00:04:24,819 --> 00:04:25,009 Well. 80 00:04:25,009 --> 00:04:26,589 What's 4 minus 10? 81 00:04:26,589 --> 00:04:27,629 Well, you're going to get a negative number. 82 00:04:27,629 --> 00:04:30,689 This is 3 to the negative sixth power. 83 00:04:30,689 --> 00:04:33,350 So using the property we just saw, you'd get 3 to the 84 00:04:33,350 --> 00:04:34,670 negative sixth power. 85 00:04:34,670 --> 00:04:37,400 Just multiplying them out, you get 1 over 3 86 00:04:37,399 --> 00:04:38,679 to the sixth power. 87 00:04:38,680 --> 00:04:41,689 And the fun part about all of this is these 88 00:04:41,689 --> 00:04:43,100 are the same quantity. 89 00:04:43,100 --> 00:04:44,930 So now you're learning a little bit about what it means 90 00:04:44,930 --> 00:04:46,519 to take a negative exponent. 91 00:04:46,519 --> 00:04:52,039 3 to the negative sixth power is equal to 1 over 3 to the 92 00:04:52,040 --> 00:04:52,629 sixth power. 93 00:04:52,629 --> 00:04:54,990 And I'm going do many, many more examples of this in the 94 00:04:54,990 --> 00:04:55,790 next video. 95 00:04:55,790 --> 00:04:58,520 But if you take anything to the negative power, so a to 96 00:04:58,519 --> 00:05:04,109 the negative b power is equal to 1 over a to the b. 97 00:05:04,110 --> 00:05:07,230 That's one thing that we just established just now. 98 00:05:07,230 --> 00:05:11,770 And earlier in this video, we saw that if I have a to the b 99 00:05:11,769 --> 00:05:17,669 over a to the c, that this is equal to a to the b minus c. 100 00:05:17,670 --> 00:05:21,009 That's the other property we've been using. 101 00:05:21,009 --> 00:05:24,509 Now, using what we've just learned and what we learned in 102 00:05:24,509 --> 00:05:32,159 the last video, let's do some more complicated problems. 103 00:05:32,160 --> 00:05:37,630 Let's say I have a to the third, b to the fourth power 104 00:05:37,629 --> 00:05:44,180 over a squared b, and all of that to the third power. 105 00:05:44,180 --> 00:05:47,290 Well, we can use the property we just learned to simplify 106 00:05:47,290 --> 00:05:48,010 the inside. 107 00:05:48,009 --> 00:05:50,800 This is going to be equal to-- a to the third 108 00:05:50,800 --> 00:05:52,650 divided by a squared. 109 00:05:52,649 --> 00:05:56,109 That's a to the 3 minus 2 power, right? 110 00:05:56,110 --> 00:05:59,189 So this would simplify to just an a. 111 00:05:59,189 --> 00:06:01,149 And you could imagine, this is a times a times a 112 00:06:01,149 --> 00:06:02,689 divided by a times a. 113 00:06:02,689 --> 00:06:04,259 You'll just have an a on top. 114 00:06:04,259 --> 00:06:09,079 And then the b, b to the fourth divided by b, well, 115 00:06:09,079 --> 00:06:12,629 that's just going to be b to the third, right? 116 00:06:12,629 --> 00:06:13,990 This is b to the first power. 117 00:06:13,990 --> 00:06:18,810 4 minus 1 is 3, and then all of that in parentheses to the 118 00:06:18,810 --> 00:06:20,139 third power. 119 00:06:20,139 --> 00:06:22,060 We don't want to forget about this third power out here. 120 00:06:22,060 --> 00:06:24,220 This third power is this one. 121 00:06:24,220 --> 00:06:25,110 Let me color code it. 122 00:06:25,110 --> 00:06:28,485 That third power is that one right there, and then this a 123 00:06:28,485 --> 00:06:30,990 in orange is that a right there. 124 00:06:30,990 --> 00:06:33,280 I think we understand what maps to what. 125 00:06:33,279 --> 00:06:35,849 And now we can use the property that when we multiply 126 00:06:35,850 --> 00:06:41,730 something and take it to the third power, this is equal to 127 00:06:41,730 --> 00:06:48,819 a to the third power times b to the third 128 00:06:48,819 --> 00:06:52,139 to the third power. 129 00:06:52,139 --> 00:06:57,699 And then this is going to be equal to a to the third power. 130 00:06:57,699 --> 00:07:03,769 131 00:07:03,769 --> 00:07:09,879 times b to the 3 times 3 power, times b to the ninth. 132 00:07:09,879 --> 00:07:11,829 And we would have simplified this about as 133 00:07:11,829 --> 00:07:14,680 far as you can go. 134 00:07:14,680 --> 00:07:16,240 Let's do one more of these. 135 00:07:16,240 --> 00:07:18,980 I think they're good practice and super-valuable 136 00:07:18,980 --> 00:07:21,150 experience later on. 137 00:07:21,149 --> 00:07:33,219 Let's say I have 25xy to the sixth over 20y 138 00:07:33,220 --> 00:07:40,190 to the fifth x squared. 139 00:07:40,189 --> 00:07:42,314 So once again, we can rearrange the numerators and 140 00:07:42,314 --> 00:07:43,399 the denominators. 141 00:07:43,399 --> 00:07:50,889 So this you could rewrite as 25 over 20 times x over x 142 00:07:50,889 --> 00:07:52,939 squared, right? 143 00:07:52,939 --> 00:07:56,170 We could have made this bottom 20x squared y to the fifth-- 144 00:07:56,170 --> 00:07:59,850 it doesn't matter the order we do it in-- times y to the 145 00:07:59,850 --> 00:08:03,140 sixth over y to the fifth. 146 00:08:03,139 --> 00:08:06,469 And let's use our newly learned exponent properties in 147 00:08:06,470 --> 00:08:08,150 actually just simplify fractions. 148 00:08:08,149 --> 00:08:13,219 25 over 20, if you divide them both by 5, this is 149 00:08:13,220 --> 00:08:17,240 equal to 5 over 4. 150 00:08:17,240 --> 00:08:20,990 x divided by x squared-- well, there's two ways you could 151 00:08:20,990 --> 00:08:21,879 think about it. 152 00:08:21,879 --> 00:08:24,889 That you could view as x to the negative 1. 153 00:08:24,889 --> 00:08:26,079 You have a first power here. 154 00:08:26,079 --> 00:08:28,240 1 minus 2 is negative 1. 155 00:08:28,240 --> 00:08:34,168 So this right here is equal to x to the negative 1 power. 156 00:08:34,168 --> 00:08:36,168 Or it could also be equal to 1 over x. 157 00:08:36,168 --> 00:08:37,620 These are equivalent. 158 00:08:37,620 --> 00:08:40,690 So let's say that this is equal into 1 over 159 00:08:40,690 --> 00:08:42,510 x, just like that. 160 00:08:42,509 --> 00:08:44,899 And it would be. x over x times x. 161 00:08:44,899 --> 00:08:47,049 One of those sets of x's would cancel out and you're just 162 00:08:47,049 --> 00:08:48,559 left with 1 over x. 163 00:08:48,559 --> 00:08:52,159 And then finally, y to the sixth over y to the fifth, 164 00:08:52,159 --> 00:08:57,539 that's y to the 6 minus 5 power, which is just y to the 165 00:08:57,539 --> 00:09:02,230 first power, or just y, so times y. 166 00:09:02,230 --> 00:09:04,730 So if you want to write it all out as just one combined 167 00:09:04,730 --> 00:09:08,409 rational expression, you have 5 times 1 times y, which would 168 00:09:08,409 --> 00:09:13,689 be 5y, all of that over 4 times x, right? 169 00:09:13,690 --> 00:09:18,930 This is y over 1, so 4 times x times 1, all of that over 4x, 170 00:09:18,929 --> 00:09:22,239 and we have successfully simplified it.