1 00:00:05,000 --> 00:00:11,340 We talked Tuesday about dividing radicals. 2 00:00:12,380 --> 00:00:13,460 We didn't really divide. 3 00:00:13,800 --> 00:00:14,700 What did we do? 4 00:00:15,180 --> 00:00:16,080 Multiply. 5 00:00:16,640 --> 00:00:17,500 We multiplied. 6 00:00:17,500 --> 00:00:20,220 and the whole goal was somebody said it, 7 00:00:21,880 --> 00:00:23,360 Rationalize the denominator. 8 00:00:23,960 --> 00:00:25,960 So square root of two is an irrational number. 9 00:00:27,600 --> 00:00:30,120 And so what are we gonna multiply by? 10 00:00:31,720 --> 00:00:34,460 Square root two over square root two. 11 00:00:35,760 --> 00:00:39,380 Yes, square root three times square root two over two. 12 00:00:40,400 --> 00:00:41,300 And we're done. 13 00:00:41,600 --> 00:00:43,060 We rationalize the denominator. 14 00:00:43,460 --> 00:00:44,820 Simplify, nothing else simplifies. 15 00:00:45,920 --> 00:00:46,760 Easy peasy, right? 16 00:00:46,760 --> 00:00:49,560 Is the cube root of two irrational? 17 00:00:51,460 --> 00:00:57,700 Yes. Unless the radicand is a perfect cube, it is irrational. 18 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 19 00:01:08,680 --> 00:01:09,380 root four. 20 00:01:09,380 --> 00:01:14,020 root of 2 over the cube root of 4. 21 00:01:16,920 --> 00:01:19,420 Is the cube root of 4 rational? 22 00:01:20,880 --> 00:01:21,780 No. 23 00:01:23,180 --> 00:01:24,100 So now what? 24 00:01:28,120 --> 00:01:31,120 Okay, I like that. 25 00:01:37,480 --> 00:01:44,100 Now we get cube root four over cube root eight. 26 00:01:44,400 --> 00:01:45,740 And what is the cube root of eight? 27 00:01:48,080 --> 00:01:50,700 Now we have a rational denominator. 28 00:01:51,460 --> 00:01:53,260 That would be the answer. 29 00:01:55,260 --> 00:01:58,380 We could have done it all in one step, instead of doing it twice. 30 00:02:09,220 --> 00:02:11,140 What should we have multiplied by? 31 00:02:12,200 --> 00:02:14,160 The cube root of four. 32 00:02:18,720 --> 00:02:22,640 Also known as the cube root of two squared. 33 00:02:27,560 --> 00:02:30,460 That gives us the cube root of two cubed. 34 00:02:30,460 --> 00:02:37,460 The result is cube root four over two. 35 00:02:37,460 --> 00:02:39,840 Let's try another one. 36 00:02:44,800 --> 00:02:45,760 No. 37 00:02:45,760 --> 00:02:49,960 Oh, let's try one. 38 00:03:00,880 --> 00:03:02,160 The denominator also contains five squared. 39 00:03:03,080 --> 00:03:06,940 So what am I gonna multiply here? 40 00:03:16,680 --> 00:03:18,660 Just the cube root of five. 41 00:03:20,940 --> 00:03:24,760 The denominator becomes the cube root of five cubed. 42 00:03:26,040 --> 00:03:28,180 Oops, still got a cube root here. 43 00:03:31,080 --> 00:03:33,300 The cube root of five cubed is just five. 44 00:03:33,300 --> 00:03:40,940 Let's try another one. 45 00:03:40,940 --> 00:03:42,340 So 46 00:03:42,340 --> 00:03:50,920 For the seventh root of two cubed, what do we need? 47 00:03:50,920 --> 00:04:05,620 We need two to the fourth power. 48 00:04:05,620 --> 00:04:12,460 Two cubed times two to the fourth is two to the seventh. 49 00:04:12,460 --> 00:04:26,940 The seventh root of two to the seventh is two. 50 00:04:26,940 --> 00:04:32,820 We add these exponents to reach the root index. 51 00:04:33,020 --> 00:04:36,640 So we choose the fourth power of that base. 52 00:04:36,640 --> 00:04:44,400 We could write this as the tenth root of four cubed. 53 00:04:56,080 --> 00:04:59,280 What power of four do we need this time? 54 00:05:00,200 --> 00:05:01,100 Seven tenths. 55 00:05:10,520 --> 00:05:14,600 And that's for the tenth root four to the tenth 56 00:05:20,180 --> 00:05:26,500 We do not need radical notation; we can keep fractional exponents. 57 00:05:30,400 --> 00:05:33,780 What power adds to three tenths to get one? 58 00:05:36,280 --> 00:05:38,840 Seven tenths. 59 00:05:40,020 --> 00:05:42,260 Multiply by four to the seven tenths over itself. 60 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. 61 00:05:52,740 --> 00:05:54,860 If you wanted to write it in radical form, you could. 62 00:05:55,020 --> 00:05:57,540 In radical form, the numerator is the tenth root of four to the seventh power. 63 00:05:58,300 --> 00:05:58,760 over four. 64 00:05:58,760 --> 00:06:04,600 We could try to reach the first power, but that would require negative five thirds. 65 00:06:04,600 --> 00:06:05,740 five thirds? 66 00:06:11,440 --> 00:06:15,820 That would leave a negative exponent in the numerator, which we do not want. 67 00:06:16,980 --> 00:06:19,680 So we fill the denominator exponent to the next whole number. 68 00:06:20,860 --> 00:06:21,820 Eight divided by three is what? 69 00:06:22,700 --> 00:06:23,740 Two and two thirds. 70 00:06:24,580 --> 00:06:26,040 The next whole exponent is three. 71 00:06:29,740 --> 00:06:33,240 So we need two to the what power? 72 00:06:37,320 --> 00:06:38,220 One third. 73 00:06:38,220 --> 00:06:54,200 Eight thirds plus one third is nine thirds, or three. 74 00:06:59,280 --> 00:07:04,080 The numerator can be written as the cube root of two. 75 00:07:06,460 --> 00:07:07,360 Like that. 76 00:07:08,100 --> 00:07:09,100 Or you could leave it like that. 77 00:07:09,300 --> 00:07:10,200 Did I write that incorrectly? 78 00:07:12,040 --> 00:07:13,060 Sorry, that exponent is three. 79 00:07:13,240 --> 00:07:13,640 Thank you. 80 00:07:13,640 --> 00:07:16,940 Thirty sevenths is four and something. 81 00:07:18,020 --> 00:07:27,340 The next whole exponent is five, or thirty-five sevenths. 82 00:07:29,420 --> 00:07:32,400 So we multiply by three to a power. 83 00:07:35,340 --> 00:07:38,240 To get from thirty to thirty-five, we need five sevenths. 84 00:07:47,680 --> 00:07:52,760 The result is three to the five sevenths over three to the fifth. 85 00:07:55,100 --> 00:07:57,960 If you did want to write it in radical form, it would be what? 86 00:07:58,320 --> 00:08:00,600 In radical form, the numerator is the seventh root of three to the fifth. 87 00:08:03,600 --> 00:08:05,040 Whatever number that comes out to be. 88 00:08:05,120 --> 00:08:05,980 Or we can leave it like that. 89 00:08:05,980 --> 00:08:06,880 Let's make it x. 90 00:08:07,920 --> 00:08:08,820 Okay. 91 00:08:10,400 --> 00:08:12,200 Did it matter what this number was? 92 00:08:14,300 --> 00:08:15,200 No. 93 00:08:16,280 --> 00:08:19,640 We use the exact same method with variables. 94 00:08:19,640 --> 00:08:21,560 What power of x do we need? 95 00:08:22,840 --> 00:08:32,540 Three fifths. Two fifths plus three fifths is five fifths. 96 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 97 00:08:39,640 --> 00:08:40,540 all. 98 00:08:41,260 --> 00:08:46,840 First, we do not want negative exponents. 99 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 100 00:08:57,204 --> 00:08:57,780 fifths. 101 00:08:57,780 --> 00:09:02,800 Then rationalize so there are no fractional exponents in the denominator. 102 00:09:02,800 --> 00:09:15,320 it's kind of the same thing as rationalizing the denominator let's try 103 00:09:15,320 --> 00:09:21,000 We want no negative exponents and no fractional exponents in the denominator. 104 00:09:21,860 --> 00:09:30,600 First rewrite the negative exponent as a reciprocal. 105 00:09:30,600 --> 00:09:39,031 That gives one over x to the positive eleven fourths; then remove the fractional 106 00:09:39,031 --> 00:09:41,440 exponent from the denominator. 107 00:09:50,680 --> 00:09:51,580 So how much do we need? 108 00:09:52,420 --> 00:09:53,320 One fourth. 109 00:09:59,120 --> 00:10:04,480 Eleven fourths plus one fourth is twelve fourths, or three. 110 00:10:05,060 --> 00:10:07,080 We will just like that. 111 00:10:09,060 --> 00:10:10,840 There are no negative exponents 112 00:10:11,660 --> 00:10:14,320 and no fractional exponents in the denominator. 113 00:10:17,220 --> 00:10:18,120 Cool? 114 00:10:18,600 --> 00:10:19,280 Okay. 115 00:10:19,280 --> 00:10:24,960 That was the last lesson before our test early next week.