WEBVTT 00:21.600 --> 00:30.264 In this lesson, we will extend our properties of exponents to include rational exponents. 00:30.320 --> 00:42.676 So let's look at the definition of a to the one divided by n. If n is an integer greater than one, 00:42.720 --> 00:53.721 and the nth root of a is a real number, then we can say that a to the one over n 00:53.760 --> 01:02.763 is equal to the nth root of a. Let's try to make this a little bit easier for you to understand. 01:02.800 --> 01:13.774 So let's suppose that we let x equal five to the one-half power. Then x squared becomes 01:13.808 --> 01:23.717 five to the one half squared. And we know multiplying one half times two gives us one. 01:23.751 --> 01:36.664 Therefore, our answer is five to the first, or just five. So if x squared is equal to five, 01:36.720 --> 01:48.676 then we know that x is equal to the square root of five. Therefore, we know that the 01:48.720 --> 01:58.519 square root of five is equal to x, and also five to the one-half power is equal to x. 01:58.560 --> 02:08.629 So the square root of five is equal to five to the one-half power. Now let's look at this. 02:08.662 --> 02:18.138 Let's suppose that x is equal to five to the one third power. Then x cubed would equal 02:18.172 --> 02:32.586 five to the one third raised to the third power, which is just five. So x cubed is equal to five. 02:32.640 --> 02:43.840 Which means x is equal to the cube root of five. And if x is equal to the cube root of x and 02:43.845 --> 02:51.572 x is equal to five to the one-third, then that means the cube root of five is equal 02:51.605 --> 03:03.784 to five to the one-third. Now maybe what you've noticed here is that the denominator 03:03.817 --> 03:15.195 of this rational exponent is the index of the radical expression. The numerator of this 03:15.229 --> 03:27.374 rational exponent is the exponent to the radicand. Let's see if the same holds true here. 03:27.440 --> 03:39.219 Denominator of two index of two, numerator of one exponent of one. 03:39.280 --> 03:46.527 Now let's work through a few examples. We're going to rewrite each of these expressions 03:46.560 --> 03:55.602 using a radical, and then we're going to evaluate them. One hundred to the 03:55.636 --> 04:08.449 one-half power is equal to, again denominator of two index of two, which is not necessary 04:08.482 --> 04:18.759 to write. One hundred to the first power. Now we need to evaluate the square root of a 04:18.792 --> 04:30.504 hundred, which we know is the square root of ten squared, which is ten. 04:30.560 --> 04:40.914 Now let's look at sixty-four to the one-third power. My denominator is three. So to rewrite 04:40.948 --> 04:52.526 this using a radical, I'm now looking at the cube root of sixty-four to the first power. 04:52.560 --> 05:05.239 I can rewrite the cube root of sixty-four, as the cube root of four cubed, which is equal to four. 05:05.280 --> 05:14.414 Now I want you to notice the difference between this example and this last one, 05:14.448 --> 05:29.229 the difference being the parentheses. In this example, only x is being raised to the one-fifth power. 05:29.280 --> 05:37.271 However in this example, the entire expression two hundred and forty-three x 05:37.304 --> 05:47.214 is being raised to the one-fifth power. So because only x is being raised to the one-fifth power 05:47.247 --> 05:56.557 in this example, then two hundred and forty-three remains two hundred and forty-three. 05:56.590 --> 06:11.238 And then I can write times the fifth root of x to the first. And that's your answer two hundred 06:11.271 --> 06:23.383 and forty-three times the fifth root of x. Now moving on to our last example. Since both two 06:23.417 --> 06:32.426 hundred and forty-three and x are raised to the one-fifth power, then we can rewrite this as 06:32.459 --> 06:45.072 the fifth root of two hundred and forty-three times x. We can then evaluate the fifth root of two hundred 06:45.105 --> 07:01.221 and forty-three. Two hundred and forty-three is three to the fifth power times x. And now to 07:01.254 --> 07:17.537 evaluate the fifth root of three to the fifth power. I get three, and I still have the fifth root of x. 07:17.600 --> 07:24.560 We'll continue our study of rational exponents by looking at some rational exponents whose 07:24.578 --> 07:33.353 numerators are numbers other than one. Let's begin with the definition of a to the m 07:33.387 --> 07:45.632 divided by n. If m and n are integers greater than one with m divided by n in lowest terms, 07:45.680 --> 07:56.276 then a to the m divided by n power is equal to the nth root of a to the m which is equal to 07:56.320 --> 08:06.286 the nth root of a raised to the m power. Let's look at an example here with numbers. 08:06.319 --> 08:15.095 So a to the two thirds, we can rewrite a to the two-thirds as a to the one-third 08:15.128 --> 08:25.600 squared. Recall multiplying one-third times two in fact gives us two-thirds. Now 08:25.605 --> 08:33.880 to rewrite this using a radical. Our denominator is the index which is three, 08:33.920 --> 08:44.191 and this is a to the second power. But we could also write a to the two-thirds another way. 08:44.240 --> 08:52.199 a to the two-thirds could also be a squared raised to the one-third power 08:52.240 --> 09:02.400 again two times one third is two-thirds, and a to the two-thirds can be written as the cube root 09:02.409 --> 09:10.320 of a squared. So now we'll take a look at a few examples. We're going to rewrite each of 09:10.350 --> 09:18.658 these expressions using a radical. And then if possible, we will evaluate that radical. 09:18.720 --> 09:26.500 Let's begin with negative one hundred and twenty-five to the two-thirds power. 09:26.560 --> 09:37.143 I can rewrite negative one hundred and twenty-five to the two-thirds power as the cube root 09:37.200 --> 09:47.220 of negative one hundred and twenty-five to the second power. I can then rewrite that 09:47.254 --> 09:56.596 as the cube root of negative one hundred and twenty-five squared. 09:56.640 --> 10:02.602 Now we can evaluate the cube root of one of negative one hundred and twenty-five. 10:02.640 --> 10:13.179 I know that negative one hundred and twenty-five can be written as the cube root of negative five 10:13.213 --> 10:29.396 cubed. And we're squaring that. Well the cube root of negative five cubed is negative five. 10:29.440 --> 10:42.075 And when I square negative five, I get twenty-five. In our second example, 10:42.108 --> 10:50.750 we're writing the number, and we're twenty-five raising it to the three-halves power. 10:50.800 --> 10:59.526 To write this expression using a radical, we're going to write it as the square root of 10:59.559 --> 11:09.469 twenty-five cubed. We can evaluate the square root of twenty-five. 11:09.520 --> 11:18.445 We know that the square root of twenty-five is the square root of five squared, 11:18.480 --> 11:29.255 and then raised to the third power. The square root of five squared is five. And five to 11:29.289 --> 11:43.937 the third power is one hundred twenty-five. Our third example is n to the four-fifths. 11:43.970 --> 11:52.912 Rewriting this expression using a radical, I'm going to get the fifth root 11:52.946 --> 12:06.760 of n to the fourth power. We can't evaluate that, so this will be our final answer. 12:06.800 --> 12:19.005 Notice in our last example the parentheses around to around ten x. That means the expression ten x 12:19.040 --> 12:34.521 is raised to the two-sevenths power. To rewrite this using a radical, it's going to look like the seventh root 12:34.560 --> 12:50.770 of ten x squared. If we square ten x, we're going to square ten which is one hundred. 12:50.804 --> 13:09.956 And squaring x will give us x squared. This cannot be evaluated any further, therefore it is our final answer. 13:10.000 --> 13:17.797 Now that you're familiar with a to the m divided by n, let's take a look at what a 13:17.840 --> 13:32.445 to the negative m divided by n may look like. a to the negative m over n is equal to one 13:32.478 --> 13:50.363 divided by a to the positive m divided by n, as long as a to the m over n is a non-zero real number. 13:50.400 --> 14:01.140 So let's rewrite each of these expressions using a radical, and then we will evaluate. 14:01.200 --> 14:11.784 Our first one is sixteen to the negative one-fourth power. I'm going to rewrite this 14:11.840 --> 14:25.680 as one divided by sixteen to the positive one-fourth power. That's our definition of a 14:25.680 --> 14:35.074 to the negative m divided by n. Now let's focus on writing sixteen to the one-fourth power 14:35.108 --> 14:50.356 using a radical. This becomes one divided by the fourth root of sixteen to the first power. 14:50.400 --> 15:07.373 I know that sixteen can be written as two to the fourth power. And then to finish evaluating 15:07.407 --> 15:21.788 this, I know that the fourth root of two to the fourth is just two. So our answer is one half. 15:21.840 --> 15:29.062 Now in our second example, we have nine to the negative five halves power. 15:29.120 --> 15:35.234 Again we're going to use our definition of a to the negative m over n 15:35.280 --> 15:46.112 to rewrite this as one divided by nine to the positive five halves power. 15:46.160 --> 15:54.687 Now we'll focus on rewriting nine to the five halves using a radical. 15:54.720 --> 16:14.607 This will look like the square root of nine to the fifth power. I know that the square root of nine is three. 16:14.640 --> 16:25.952 So one divided by three to the fifth which is equal to one divided by two hundred and forty-three. 16:26.000 --> 16:37.196 And that's the answer to that problem. Now we're going to use our rules for exponents 16:37.230 --> 16:53.779 to simplify expressions. Here is a summary of exponent rules, all of which we have studied before. 16:53.840 --> 17:01.387 In our first example, we're going to multiply, and we want to assume that all variables 17:01.420 --> 17:10.429 represent positive real numbers. So here we're taking b to the one third, and we're multiplying 17:10.463 --> 17:21.774 it times b to the eleven-thirds. So here we are multiplying like bases, and so we know 17:21.807 --> 17:32.451 that we're going to add our exponents. We can write this as b to the one-third plus 17:32.485 --> 17:43.563 eleven-thirds, which we know is b to the twelve thirds. And twelve divided by three is four. 17:43.596 --> 17:56.075 So this is equal to b to the fourth power. Now in our second example, what we're doing here is 17:56.108 --> 18:05.918 we're multiplying x to the one-fourth times x to the eighth minus three x. So again, we're going 18:05.952 --> 18:13.926 to be adding our exponents. So x to the one-fourth times x to the one, I'm sorry, 18:13.960 --> 18:26.272 x to the eighth is going to give us x to the one-fourth plus eight minus, 18:26.320 --> 18:31.143 now we're taking x to the one-fourth and we're multiplying it times 18:31.200 --> 18:42.388 three x. That's going to give us three x. Now remember this is x to the first power. 18:42.400 --> 18:52.465 There is an understood one as the exponent to x here. So x to the one-fourth times x to the first 18:52.498 --> 19:02.375 is going to give us x to the one-fourth plus one. Now we can add these numbers 19:02.408 --> 19:09.648 together in the exponents position. And that's going to give us one-fourth plus 19:09.682 --> 19:24.530 eight. Gives us x to the thirty-three fourths. Minus three x one-fourth plus one gives us 19:24.563 --> 19:35.007 five-fourths, and this is our final answer to this example. 19:35.040 --> 19:43.149 Let's look at our next example, we're going to use rational exponents to write each of 19:43.182 --> 19:50.623 these as a single radical expression. And we are to assume that all variables represent 19:50.656 --> 19:57.163 positive real numbers. So in my first example I have the cube root of t 19:57.200 --> 20:04.170 times the fourth root of t. So first we're going to rewrite each of these using 20:04.203 --> 20:15.080 rational exponents. I can write the cube root of t, now remember this is understood t 20:15.120 --> 20:25.191 to the first power. So the cube root of t, I can write as t to the one third. Again the 20:25.224 --> 20:33.732 numerator is the exponent here, and the denominator is your index. Now to write 20:33.766 --> 20:45.778 the fourth root of t using a rational exponent. That's going to look like t to the one-fourth. 20:45.840 --> 20:52.551 And now what we have here is t to the one third times t to the one-fourth, which 20:52.585 --> 21:00.359 remember multiplying like bases, we're going to add the exponents. This will give 21:00.392 --> 21:14.506 me t to the one-third plus one-fourth. And that is equal to t to the seven twelfths. 21:14.540 --> 21:27.553 Now let's take t to the seven-twelfths and rewrite that using a radical. 21:27.600 --> 21:34.593 Your denominator is the index, so this will look like the twelfth root of t 21:34.627 --> 21:46.906 to the seventh power. Now in our next example, you'll notice we are dividing. 21:46.960 --> 21:55.948 But first let's rewrite each of these radical expressions using a rational exponent. 21:56.000 --> 22:05.157 The cube root of t, I can write as t to the one-third. Divided by the fourth root of t, 22:05.190 --> 22:15.334 looks like t to the one fourth. Now in this case we are dividing like bases. And when we 22:15.367 --> 22:27.212 divide like bases, we subtract the exponents. Subtracting my exponents will give me t 22:27.246 --> 22:44.263 to the one-third minus one-fourth. t to the one-third minus one-fourth is t to the one-twelfth. 22:44.296 --> 22:53.372 And now we will take t to the one-twelfth, and re-write is using a radical. That will 22:53.405 --> 23:05.517 look like the twelfth root of t to the first power which is just the twelfth root of t. 23:05.551 --> 23:16.061 Let's continue our practice of working with rational exponents. We're going to use the properties 23:16.095 --> 23:23.001 of exponents to simplify each of these expressions. We're going to write them with 23:23.035 --> 23:29.608 positive exponents. And we are to assume that all variables represent positive real 23:29.641 --> 23:39.118 numbers. So in our first example, we have four u to the negative two raised to the three- 23:39.151 --> 23:49.695 halves power. So recall that when we have a base raised to an exponent and raised to 23:49.728 --> 24:02.407 another exponent, we're multiplying those exponents. So we can rewrite this as four to 24:02.441 --> 24:18.790 the three-halves power times u to the negative two times three-halves. Four to the 24:18.824 --> 24:29.134 three-halves power, we can write using a radical as the square root of four 24:29.200 --> 24:41.847 to the third power times negative two times three-halves will give us negative three. So u 24:41.880 --> 24:51.523 to the negative two times three-halves is the same as u to the negative three. Now we 24:51.557 --> 25:06.004 know that the square root of four can be written as two squared. And since our 25:06.038 --> 25:13.011 directions say to write using positive exponents. Then I can rewrite u to the 25:13.045 --> 25:24.223 negative third power, as one divided by u to the positive third power. Now the square 25:24.256 --> 25:39.972 root of two squared is just two. And two cubed is eight. Eight times one divided by 25:40.005 --> 25:52.751 u cubed will give us eight divided by u cubed as our final answer. 25:52.800 --> 26:00.640 Looking at our second example, I notice in my numerator we're multiplying like bases, 26:00.659 --> 26:09.334 which means we're going to add our exponents. Let's do that first. m to the three-fifths 26:09.368 --> 26:19.011 times m to the one-fifth will look like m to the three-fifths plus one-fifth. And we're 26:19.044 --> 26:32.524 dividing that by m to the negative one fifth. Three-fifths plus one-fifth is four-fifths. 26:32.560 --> 26:43.635 So m to the three-fifths plus one-fifth is m to the four-fifths. And we are 26:43.668 --> 26:54.080 dividing that by m to the negative one-fifth. Remember when we divide like bases, 26:54.080 --> 27:04.823 we subtract our exponents. So m to the four-fifths divided by m to the negative one-fifth 27:04.880 --> 27:14.066 can be written as m to the four-fifths minus negative one-fifth. And we know that 27:14.099 --> 27:22.107 subtracting negative one-fifth is the same as adding one-fifth, so this is equal to m to 27:22.140 --> 27:33.485 the four-fifths plus one-fifth, which is m to the five-fifths. And five divided by five is one. 27:33.518 --> 27:47.840 So this is equal to m to the first power, or just m.