WEBVTT 00:22.720 --> 00:29.563 In this section, we're going to discuss negative exponents. In previous sections, we have 00:29.600 --> 00:35.536 used and learned rules for whole number exponents, and we're now going to extend our understanding to 00:35.600 --> 00:44.845 include negative integer exponents. We'll begin with an exploration of x squared divided by x 00:44.880 --> 00:52.920 to the fifth. Let's consider this expression x squared divided by x to the fifth and simplify 00:52.960 --> 00:59.326 it using the quotient rule. We've learned previously that when I divide an expression 00:59.360 --> 01:06.500 by another expression with the same base that I will keep the base and subtract the exponents. 01:06.560 --> 01:16.911 So this would give me x to the two minus five, or x to the negative three. 01:16.960 --> 01:24.985 Now considering the same expression x squared divided by x to the fifth, 01:25.040 --> 01:34.428 I know that I can write this expression as x times x in the numerator and then 01:34.480 --> 01:45.472 x times itself five times in the denominator to represent x to the fifth. 01:45.520 --> 01:51.478 Using the fundamental principle of fractions, we can reduce this fraction by factoring out the 01:51.520 --> 02:01.221 like factors. I know that x divided by itself is just one, so the common factors in this expression 02:01.280 --> 02:18.205 is an x squared. This will leave us when I simplify this as one over x times x times x, which is x to the third power. 02:18.240 --> 02:27.381 Well, since we know that x squared over x to the fifth is equal to x to the negative three, 02:27.440 --> 02:37.224 and it is also equal to one over x cubed. Then that means that x to the negative three must be 02:37.280 --> 02:57.378 equal to one over x cubed. This leads us to a principle for negative exponents. 02:57.440 --> 03:15.796 We can say that a to the negative n is equal to one over a to the n for values of a that are not zero and for n's that are integers. 03:15.840 --> 03:23.937 Let's look at a few examples. We begin with six to the negative two power. The negative exponent 03:24.000 --> 03:33.547 means that we can rewrite this expression as one divided by six to the positive two power. 03:33.600 --> 03:43.957 And six to the positive two power, also called six squared, is thirty-six. So this gives us one over 03:44.000 --> 03:57.704 thirty-six. Keep in mind that a negative exponent does not change the sign of the base in any way. 03:57.760 --> 04:09.183 In our next example, we have negative three raised to the negative four. We can rewrite this as one divided by 04:09.216 --> 04:21.161 negative three to the positive four. Rewriting our exponent from a negative exponent to a positive did not change our base. 04:21.200 --> 04:28.435 Now we need to evaluate negative three to the fourth power, which is negative three times 04:28.480 --> 04:42.583 itself four times. This will give us a positive eighty-one, resulting here in one over eighty-one. 04:42.640 --> 04:50.123 And in our last example, we have one-half raised to the negative three power. 04:50.160 --> 04:59.866 This gives us one divided by one-half to the positive three power. 04:59.920 --> 05:08.108 Now recalling our rules for exponents of quotients, this three can be applied to both 05:08.160 --> 05:21.221 the one and the two in the fraction of one-half, giving us one divided by one cubed over two cubed. 05:21.280 --> 05:27.294 We know that dividing by a fraction is the same as multiplying by a reciprocal, 05:27.360 --> 05:36.103 so we can rewrite this again as one times the reciprocal of one cubed over two cubed, 05:36.136 --> 05:54.421 which would give us two cubed over one cubed, or eight divided by one, which is eight. 05:54.480 --> 06:02.429 All of the previously stated rules for exponents apply also for negative exponents. Let's look at 06:02.480 --> 06:09.703 these next few problems which will combine all of the exponent rules that we've learned so far. 06:09.760 --> 06:16.510 We want to simplify the expression by rewriting using only positive exponents. 06:16.560 --> 06:22.916 We begin with four a to the negative nine. Let's begin by noticing what is it that's 06:22.960 --> 06:32.826 being raised to the exponent, negative nine. It is only the a. So the four will remain as it is, 06:32.880 --> 06:39.399 and a to the negative nine can be written as one over a to the ninth. 06:39.440 --> 06:49.643 So we see that we have four times one over a to the ninth, which is the same as four over 06:49.680 --> 07:00.420 a to the ninth. We've rewritten the problem with only a positive exponent. 07:00.480 --> 07:11.498 In the next problem, we have three divided by b to the negative five. Again, notice that b is being raised to the exponent, 07:11.531 --> 07:22.442 negative five. So the three will remain in the numerator. b to the negative five can be written as one 07:22.480 --> 07:31.151 over b to the fifth. And recall that dividing by a fraction is the same as multiplying by 07:31.200 --> 07:41.040 its reciprocal, so this will be the same as three times the reciprocal of one over b to the fifth, 07:41.061 --> 07:51.705 which is just b to the fifth, leaving us with three b to the fifth power. 07:51.760 --> 07:58.511 In our third example, we have x to the negative eight divided by x squared. 07:58.560 --> 08:04.751 Let's use our quotient rule to simplify to begin with. We have the same base 08:04.800 --> 08:12.993 on these expressions, and we're dividing them, so we subtract the exponents. 08:13.040 --> 08:23.703 We keep the base x, and we have negative eight minus two as our new exponent, giving us x 08:23.760 --> 08:30.910 to the negative ten. Well, our directions tell us to rewrite using positive exponents, 08:30.960 --> 08:43.790 so we can rewrite again. x to the negative ten is the same as one over x to the positive ten. 08:43.840 --> 08:50.897 Our next example is fifteen p to the sixth q divided by three p to the negative one 08:50.960 --> 09:01.708 q to the fourth. Let's begin by separating these factors by their bases, by their like bases. 09:01.760 --> 09:09.040 I'll start with my integers however and combine those into one expression. Fifteen 09:09.048 --> 09:19.626 divided by three can be grouped, and then p to the six divided by p to the negative one, 09:19.680 --> 09:31.404 and q to the first divided by q to the fourth. Notice I added a one as the exponent on this q. 09:31.440 --> 09:37.477 There is an implied exponent of one whenever we don't see an exponent. 09:37.520 --> 09:48.755 Now let us simplify each of these factors. First, fifteen divided by three is five. 09:48.800 --> 09:57.564 p to the sixth divided by p to the negative one is p to the six minus 09:57.600 --> 10:10.343 a negative one. And q to the first divided by q to the fourth is q to the one minus four. 10:10.400 --> 10:27.060 This gives us five p to the six plus one, or seventh, power q to the negative three power. 10:27.120 --> 10:33.800 And in our final step because we want to rewrite this expression with only positive exponents, 10:33.840 --> 10:39.939 we notice that we have a q to the negative three which is the same as one over q to 10:40.000 --> 10:49.716 the third power. So five p to the seventh will remain in the numerator of our answer, 10:49.760 --> 10:59.225 and q to the third power will be in the denominator of our answer. 10:59.280 --> 11:07.000 In our next example, we have x squared y to the fourth raised to the negative two power. 11:07.040 --> 11:19.245 Here we have a product being raised to a power. So we can apply this power of negative two to each of these factors. 11:19.280 --> 11:31.057 This gives us x squared to the negative two times y to the fourth to the negative two. 11:31.120 --> 11:39.799 And now we have a power raised to a power, and we know to multiply these exponents. 11:39.840 --> 11:57.850 So this gives me x to the negative four y to the negative eight. And now as I simplify these expressions, or this expression, 11:57.884 --> 12:05.792 each of these factors are raised to a negative power which means x to the negative four will become a one 12:05.840 --> 12:16.903 over x to the fourth, putting that factor in the denominator. And y to the negative eight is the same as one over y to the eighth, 12:16.936 --> 12:30.817 meaning that y to the eighth will be in the denominator. And that leaves us with a one in the numerator. 12:30.880 --> 12:37.957 In our last example, we have two a b to the negative three times negative four 12:38.000 --> 12:46.199 a to the negative three b squared. So we're going to multiply these two factors together. 12:46.240 --> 12:51.871 We know that by the Commutative and Associative Property of Multiplication that we can rearrange 12:51.920 --> 13:03.082 these factors and regroup them. So let's do that. We will multiply two times negative four 13:03.120 --> 13:18.931 as one grouping and a times a to the negative three as another grouping and b to the negative three times b squared 13:18.965 --> 13:29.141 as our third grouping. Now let's apply our rules of exponents and multiplication. Two times negative four 13:29.200 --> 13:40.753 is negative eight. a, remember to the first power is implied, times a to the negative three will give us a to the 13:40.786 --> 13:51.664 one plus a negative three in the exponent. And then b to the negative three times b squared will give us 13:51.697 --> 14:10.016 b to the negative three plus two. This gives us negative eight a to the negative two b to the negative one. 14:10.080 --> 14:16.756 And finally, to clear our negative exponents, we recall that a to the negative two is the same as 14:16.800 --> 14:26.265 one over a squared, and b to the negative one is one over b. So these factors will 14:26.320 --> 14:43.840 move to the denominator when we write our final answer, negative eight divided by a squared b.