WEBVTT 00:00:05.000 --> 00:00:11.340 We talked Tuesday about dividing radicals. 00:00:12.380 --> 00:00:13.460 We didn't really divide. 00:00:13.800 --> 00:00:14.700 What did we do? 00:00:15.180 --> 00:00:16.080 Multiply. 00:00:16.640 --> 00:00:17.500 We multiplied. 00:00:17.500 --> 00:00:20.220 and the whole goal was somebody said it, 00:00:21.880 --> 00:00:23.360 Rationalize the denominator. 00:00:23.960 --> 00:00:25.960 So square root of two is an irrational number. 00:00:27.600 --> 00:00:30.120 And so what are we gonna multiply by? 00:00:31.720 --> 00:00:34.460 Square root two over square root two. 00:00:35.760 --> 00:00:39.380 Yes, square root three times square root two over two. 00:00:40.400 --> 00:00:41.300 And we're done. 00:00:41.600 --> 00:00:43.060 We rationalize the denominator. 00:00:43.460 --> 00:00:44.820 Simplify, nothing else simplifies. 00:00:45.920 --> 00:00:46.760 Easy peasy, right? 00:00:46.760 --> 00:00:49.560 Is the cube root of two irrational? 00:00:51.460 --> 00:00:57.700 Yes. Unless the radicand is a perfect cube, it is irrational. 00:01:02.380 --> 00:01:08.680 If I multiply by cube root two over cube root two, I get cube root two over cube 00:01:08.680 --> 00:01:09.380 root four. 00:01:09.380 --> 00:01:14.020 root of 2 over the cube root of 4. 00:01:16.920 --> 00:01:19.420 Is the cube root of 4 rational? 00:01:20.880 --> 00:01:21.780 No. 00:01:23.180 --> 00:01:24.100 So now what? 00:01:28.120 --> 00:01:31.120 Okay, I like that. 00:01:37.480 --> 00:01:44.100 Now we get cube root four over cube root eight. 00:01:44.400 --> 00:01:45.740 And what is the cube root of eight? 00:01:48.080 --> 00:01:50.700 Now we have a rational denominator. 00:01:51.460 --> 00:01:53.260 That would be the answer. 00:01:55.260 --> 00:01:58.380 We could have done it all in one step, instead of doing it twice. 00:02:09.220 --> 00:02:11.140 What should we have multiplied by? 00:02:12.200 --> 00:02:14.160 The cube root of four. 00:02:18.720 --> 00:02:22.640 Also known as the cube root of two squared. 00:02:27.560 --> 00:02:30.460 That gives us the cube root of two cubed. 00:02:30.460 --> 00:02:37.460 The result is cube root four over two. 00:02:37.460 --> 00:02:39.840 Let's try another one. 00:02:44.800 --> 00:02:45.760 No. 00:02:45.760 --> 00:02:49.960 Oh, let's try one. 00:03:00.880 --> 00:03:02.160 The denominator also contains five squared. 00:03:03.080 --> 00:03:06.940 So what am I gonna multiply here? 00:03:16.680 --> 00:03:18.660 Just the cube root of five. 00:03:20.940 --> 00:03:24.760 The denominator becomes the cube root of five cubed. 00:03:26.040 --> 00:03:28.180 Oops, still got a cube root here. 00:03:31.080 --> 00:03:33.300 The cube root of five cubed is just five. 00:03:33.300 --> 00:03:40.940 Let's try another one. 00:03:40.940 --> 00:03:42.340 So 00:03:42.340 --> 00:03:50.920 For the seventh root of two cubed, what do we need? 00:03:50.920 --> 00:04:05.620 We need two to the fourth power. 00:04:05.620 --> 00:04:12.460 Two cubed times two to the fourth is two to the seventh. 00:04:12.460 --> 00:04:26.940 The seventh root of two to the seventh is two. 00:04:26.940 --> 00:04:32.820 We add these exponents to reach the root index. 00:04:33.020 --> 00:04:36.640 So we choose the fourth power of that base. 00:04:36.640 --> 00:04:44.400 We could write this as the tenth root of four cubed. 00:04:56.080 --> 00:04:59.280 What power of four do we need this time? 00:05:00.200 --> 00:05:01.100 Seven tenths. 00:05:10.520 --> 00:05:14.600 And that's for the tenth root four to the tenth 00:05:20.180 --> 00:05:26.500 We do not need radical notation; we can keep fractional exponents. 00:05:30.400 --> 00:05:33.780 What power adds to three tenths to get one? 00:05:36.280 --> 00:05:38.840 Seven tenths. 00:05:40.020 --> 00:05:42.260 Multiply by four to the seven tenths over itself. 00:05:43.660 --> 00:05:47.980 Four to the three tenths times four to the seven tenths is four to the first power. 00:05:52.740 --> 00:05:54.860 If you wanted to write it in radical form, you could. 00:05:55.020 --> 00:05:57.540 In radical form, the numerator is the tenth root of four to the seventh power. 00:05:58.300 --> 00:05:58.760 over four. 00:05:58.760 --> 00:06:04.600 We could try to reach the first power, but that would require negative five thirds. 00:06:04.600 --> 00:06:05.740 five thirds? 00:06:11.440 --> 00:06:15.820 That would leave a negative exponent in the numerator, which we do not want. 00:06:16.980 --> 00:06:19.680 So we fill the denominator exponent to the next whole number. 00:06:20.860 --> 00:06:21.820 Eight divided by three is what? 00:06:22.700 --> 00:06:23.740 Two and two thirds. 00:06:24.580 --> 00:06:26.040 The next whole exponent is three. 00:06:29.740 --> 00:06:33.240 So we need two to the what power? 00:06:37.320 --> 00:06:38.220 One third. 00:06:38.220 --> 00:06:54.200 Eight thirds plus one third is nine thirds, or three. 00:06:59.280 --> 00:07:04.080 The numerator can be written as the cube root of two. 00:07:06.460 --> 00:07:07.360 Like that. 00:07:08.100 --> 00:07:09.100 Or you could leave it like that. 00:07:09.300 --> 00:07:10.200 Did I write that incorrectly? 00:07:12.040 --> 00:07:13.060 Sorry, that exponent is three. 00:07:13.240 --> 00:07:13.640 Thank you. 00:07:13.640 --> 00:07:16.940 Thirty sevenths is four and something. 00:07:18.020 --> 00:07:27.340 The next whole exponent is five, or thirty-five sevenths. 00:07:29.420 --> 00:07:32.400 So we multiply by three to a power. 00:07:35.340 --> 00:07:38.240 To get from thirty to thirty-five, we need five sevenths. 00:07:47.680 --> 00:07:52.760 The result is three to the five sevenths over three to the fifth. 00:07:55.100 --> 00:07:57.960 If you did want to write it in radical form, it would be what? 00:07:58.320 --> 00:08:00.600 In radical form, the numerator is the seventh root of three to the fifth. 00:08:03.600 --> 00:08:05.040 Whatever number that comes out to be. 00:08:05.120 --> 00:08:05.980 Or we can leave it like that. 00:08:05.980 --> 00:08:06.880 Let's make it x. 00:08:07.920 --> 00:08:08.820 Okay. 00:08:10.400 --> 00:08:12.200 Did it matter what this number was? 00:08:14.300 --> 00:08:15.200 No. 00:08:16.280 --> 00:08:19.640 We use the exact same method with variables. 00:08:19.640 --> 00:08:21.560 What power of x do we need? 00:08:22.840 --> 00:08:32.540 Three fifths. Two fifths plus three fifths is five fifths. 00:08:34.520 --> 00:08:39.640 So on the warm up we did the other day, I didn't really explain that very well or at 00:08:39.640 --> 00:08:40.540 all. 00:08:41.260 --> 00:08:46.840 First, we do not want negative exponents. 00:08:46.840 --> 00:08:57.204 For x to the negative two fifths, first rewrite it as one over x to the positive two 00:08:57.204 --> 00:08:57.780 fifths. 00:08:57.780 --> 00:09:02.800 Then rationalize so there are no fractional exponents in the denominator. 00:09:02.800 --> 00:09:15.320 it's kind of the same thing as rationalizing the denominator let's try 00:09:15.320 --> 00:09:21.000 We want no negative exponents and no fractional exponents in the denominator. 00:09:21.860 --> 00:09:30.600 First rewrite the negative exponent as a reciprocal. 00:09:30.600 --> 00:09:39.031 That gives one over x to the positive eleven fourths; then remove the fractional 00:09:39.031 --> 00:09:41.440 exponent from the denominator. 00:09:50.680 --> 00:09:51.580 So how much do we need? 00:09:52.420 --> 00:09:53.320 One fourth. 00:09:59.120 --> 00:10:04.480 Eleven fourths plus one fourth is twelve fourths, or three. 00:10:05.060 --> 00:10:07.080 We will just like that. 00:10:09.060 --> 00:10:10.840 There are no negative exponents 00:10:11.660 --> 00:10:14.320 and no fractional exponents in the denominator. 00:10:17.220 --> 00:10:18.120 Cool? 00:10:18.600 --> 00:10:19.280 Okay. 00:10:19.280 --> 00:10:24.960 That was the last lesson before our test early next week.