1 00:00:00,000 --> 00:00:00,560 2 00:00:00,560 --> 00:00:03,589 So far, when we were dealing with radicals we've only been 3 00:00:03,589 --> 00:00:05,370 using the square root. 4 00:00:05,370 --> 00:00:09,189 We've seen that if I write a radical sign like this and put 5 00:00:09,189 --> 00:00:13,179 a 9 under it, this means the principal square root of 9, 6 00:00:13,179 --> 00:00:14,689 which is positive 3. 7 00:00:14,689 --> 00:00:17,879 Or you could view it as the positive square root of 9. 8 00:00:17,879 --> 00:00:20,839 Now, what's implicit when we write it like this is that I'm 9 00:00:20,839 --> 00:00:23,359 taking the square root. 10 00:00:23,359 --> 00:00:25,669 So I could have also written it like this. 11 00:00:25,670 --> 00:00:27,640 I could have also written the radical sign like this and 12 00:00:27,640 --> 00:00:32,039 written this index 2 here, which means the square root, 13 00:00:32,039 --> 00:00:35,570 the principal square root of 9. 14 00:00:35,570 --> 00:00:37,679 Find me something that if I square that 15 00:00:37,679 --> 00:00:40,130 something, I get 9. 16 00:00:40,130 --> 00:00:42,330 And the radical sign doesn't just have to 17 00:00:42,329 --> 00:00:44,750 apply to a square root. 18 00:00:44,750 --> 00:00:48,210 You can change the index here and then take an arbitrary 19 00:00:48,210 --> 00:00:49,660 root of a number. 20 00:00:49,659 --> 00:00:54,390 So for example, if I were to ask you, what-- You could 21 00:00:54,390 --> 00:00:57,039 imagine this is called the cube root, or you could call 22 00:00:57,039 --> 00:01:00,780 it the third root of 27. 23 00:01:00,780 --> 00:01:01,980 What is this? 24 00:01:01,979 --> 00:01:05,119 Well, this is some number that if I take it to the third 25 00:01:05,120 --> 00:01:07,910 power, I'd get 27. 26 00:01:07,909 --> 00:01:09,420 Well, the only number that if you take it to the third 27 00:01:09,420 --> 00:01:12,540 power, you get 27 is equal to 3. 28 00:01:12,540 --> 00:01:18,420 3 times 3 times 3 is equal to 27. 29 00:01:18,420 --> 00:01:23,040 9 times 3, 27. 30 00:01:23,040 --> 00:01:25,720 So likewise, let me just do one more. 31 00:01:25,719 --> 00:01:29,760 So if I have 16-- I'll do it in a different color. 32 00:01:29,760 --> 00:01:34,780 If I have 16 and I want to take the fourth root of 16, 33 00:01:34,780 --> 00:01:38,840 what number times itself 4 times is equal to 16? 34 00:01:38,840 --> 00:01:41,030 And if it doesn't pop out at you immediately, you can 35 00:01:41,030 --> 00:01:43,920 actually just do a prime factorization of 16 36 00:01:43,920 --> 00:01:45,250 to figure it out. 37 00:01:45,250 --> 00:01:45,439 Let's see. 38 00:01:45,439 --> 00:01:47,829 16 is 2 times 8. 39 00:01:47,829 --> 00:01:50,120 8 is 2 times 4. 40 00:01:50,120 --> 00:01:52,340 4 is 2 times 2. 41 00:01:52,340 --> 00:01:58,010 So this is equal to the fourth root of 2 times 2 42 00:01:58,010 --> 00:01:59,910 times 2 times 2. 43 00:01:59,909 --> 00:02:01,509 You have these four 2's here. 44 00:02:01,510 --> 00:02:05,450 Well, I have four 2's being multiplied, so the fourth root 45 00:02:05,450 --> 00:02:10,039 of this must be equal to 2. 46 00:02:10,038 --> 00:02:11,750 And you could also view this as kind of the fourth 47 00:02:11,750 --> 00:02:15,750 principal root because if these were all negative 2's, 48 00:02:15,750 --> 00:02:17,099 it would also work. 49 00:02:17,099 --> 00:02:21,889 50 00:02:21,889 --> 00:02:24,359 Just like you have multiple square roots, you have 51 00:02:24,360 --> 00:02:25,540 multiple fourth roots. 52 00:02:25,539 --> 00:02:29,019 But the radical sign implies the principal root. 53 00:02:29,020 --> 00:02:32,530 Now, with that said, we've simplified traditional square 54 00:02:32,530 --> 00:02:33,509 roots before. 55 00:02:33,509 --> 00:02:36,939 Now we should hopefully be able to simplify radicals with 56 00:02:36,939 --> 00:02:38,155 higher power roots. 57 00:02:38,155 --> 00:02:39,979 So let's try a couple. 58 00:02:39,979 --> 00:02:42,789 Let's say I want to simplify this expression. 59 00:02:42,789 --> 00:02:47,139 The fifth root of 96. 60 00:02:47,139 --> 00:02:50,859 So like I said before, let's just factor this right here. 61 00:02:50,860 --> 00:02:55,080 So 96 is 2 times 48. 62 00:02:55,080 --> 00:02:58,000 Which is 2 times 24. 63 00:02:58,000 --> 00:03:00,639 Which is 2 times 12. 64 00:03:00,639 --> 00:03:03,159 Which is 2 times 6. 65 00:03:03,159 --> 00:03:05,969 Which is 2 times 3. 66 00:03:05,969 --> 00:03:12,789 So this is equal to the fifth root of 2 times 2 times 2 67 00:03:12,789 --> 00:03:14,039 times 2 times 2. 68 00:03:14,039 --> 00:03:20,090 69 00:03:20,090 --> 00:03:23,310 Times 3. 70 00:03:23,310 --> 00:03:26,520 Or another way you could view it, is you could view it to a 71 00:03:26,520 --> 00:03:28,540 fractional power. 72 00:03:28,539 --> 00:03:30,539 You could view it to a fractional power. 73 00:03:30,539 --> 00:03:31,719 We've talked about that already. 74 00:03:31,719 --> 00:03:36,979 This is the same thing as 2 times 2 times 2 times 2 times 75 00:03:36,979 --> 00:03:41,780 2 times 3 to the 1/5 power. 76 00:03:41,780 --> 00:03:44,330 Let me make this clear. 77 00:03:44,330 --> 00:03:49,270 Having an nth root of some number is equivalent to taking 78 00:03:49,270 --> 00:03:52,560 that number to the 1/n power. 79 00:03:52,560 --> 00:03:55,509 These are equivalent statements right here. 80 00:03:55,509 --> 00:03:57,639 So if you're taking this to the 1/5 power, this is the 81 00:03:57,639 --> 00:04:01,219 same thing as taking 2 times 2 times 2 times 2 82 00:04:01,219 --> 00:04:03,639 times 2 to the 1/5. 83 00:04:03,639 --> 00:04:06,149 Times 3 to the 1/5. 84 00:04:06,150 --> 00:04:08,760 Now I have something that's multiplied. 85 00:04:08,759 --> 00:04:11,629 I have 2 multiplied by itself 5 times. 86 00:04:11,629 --> 00:04:13,280 And I'm taking that to the 1/5. 87 00:04:13,280 --> 00:04:15,180 Well, the 1/5 power of this is going to be 2. 88 00:04:15,180 --> 00:04:17,569 Or the fifth root of this is just going to be 2. 89 00:04:17,569 --> 00:04:19,219 So this is going to be a 2 right here. 90 00:04:19,220 --> 00:04:21,899 And this is going to be 3 to the 1/5 power. 91 00:04:21,899 --> 00:04:26,860 2 times 3 to the 1/5, which is this simplified about as much 92 00:04:26,860 --> 00:04:27,750 as you can simplify it. 93 00:04:27,750 --> 00:04:29,790 But if we want to keep in radical form, we could write 94 00:04:29,790 --> 00:04:39,020 it as 2 times the fifth root 3 just like that. 95 00:04:39,019 --> 00:04:40,269 Let's try another one. 96 00:04:40,269 --> 00:04:44,319 97 00:04:44,319 --> 00:04:46,480 Let me put some variables in there. 98 00:04:46,480 --> 00:04:53,850 Let's say we wanted to simplify the sixth root of 64 99 00:04:53,850 --> 00:04:56,570 times x to the eighth. 100 00:04:56,569 --> 00:04:59,420 So let's do 64 first. 101 00:04:59,420 --> 00:05:06,230 64 is equal to 2 times 32, which is 2 times 16. 102 00:05:06,230 --> 00:05:07,980 Which is 2 times 8. 103 00:05:07,980 --> 00:05:09,610 Which is 2 times 4. 104 00:05:09,610 --> 00:05:11,860 Which is 2 times 2. 105 00:05:11,860 --> 00:05:14,730 So we have 1, 2, 3, 4, 5, 6. 106 00:05:14,730 --> 00:05:16,900 So it's essentially 2 to the sixth power. 107 00:05:16,899 --> 00:05:21,879 So this is equivalent to the sixth root of 2 to the sixth-- 108 00:05:21,879 --> 00:05:28,529 that's what 64 is --times x to the eighth power. 109 00:05:28,529 --> 00:05:31,500 Now, the sixth root of 2 to the sixth, that's pretty 110 00:05:31,500 --> 00:05:33,139 straightforward. 111 00:05:33,139 --> 00:05:39,289 So this part right here is just going to be equal to 2. 112 00:05:39,290 --> 00:05:47,460 That's going to be 2 times the sixth root of x 113 00:05:47,459 --> 00:05:48,709 to the eighth power. 114 00:05:48,709 --> 00:05:51,229 115 00:05:51,230 --> 00:05:53,300 And how can we simplify this? 116 00:05:53,300 --> 00:05:58,310 Well, x to the eighth power, that's the same thing as x to 117 00:05:58,310 --> 00:06:02,860 the sixth power times x squared. 118 00:06:02,860 --> 00:06:04,629 You have the same base, you would add the exponents. 119 00:06:04,629 --> 00:06:07,050 This is the same thing as x to the eighth. 120 00:06:07,050 --> 00:06:11,960 So this is going to be equal to 2 times the sixth root of x 121 00:06:11,959 --> 00:06:14,939 to the sixth times x squared. 122 00:06:14,939 --> 00:06:18,500 And the sixth root, this part right here, the sixth root of 123 00:06:18,500 --> 00:06:20,769 x to the sixth, that's just x. 124 00:06:20,769 --> 00:06:26,269 So this is going to be equal to 2 times x times the sixth 125 00:06:26,269 --> 00:06:30,899 root of x squared. 126 00:06:30,899 --> 00:06:33,810 Now, we can simplify this even more if you 127 00:06:33,810 --> 00:06:34,569 really think about. 128 00:06:34,569 --> 00:06:37,990 Remember, this expression right here, this is the exact 129 00:06:37,990 --> 00:06:44,129 same thing as x squared to the 1/6 power. 130 00:06:44,129 --> 00:06:47,100 And if you remember your exponent properties, when you 131 00:06:47,100 --> 00:06:49,620 raise something to an exponent, and then raise that 132 00:06:49,620 --> 00:06:53,800 to an exponent, that's equivalent to x to 133 00:06:53,800 --> 00:06:56,490 the 2 times 1/6 power. 134 00:06:56,490 --> 00:07:00,720 Or-- let me write this --2 times 1/6 power, which is the 135 00:07:00,720 --> 00:07:05,010 same thing-- Let me not forget to write my 2x there. 136 00:07:05,009 --> 00:07:08,079 So I have a 2x there and a 2x there. 137 00:07:08,079 --> 00:07:12,300 And this is the same thing as 2x-- it's the same 2x there 138 00:07:12,300 --> 00:07:16,810 --times x to the 2/6. 139 00:07:16,810 --> 00:07:20,220 Or, if we want to write that in most simple form or lowest 140 00:07:20,220 --> 00:07:26,710 common form, you get 2x times x to the-- What do you have 141 00:07:26,709 --> 00:07:28,870 here? x to the 1/3. 142 00:07:28,870 --> 00:07:31,319 So if you want to write it in radical form, you could write 143 00:07:31,319 --> 00:07:38,180 this is equal to 2 times 2x times the third root of x. 144 00:07:38,180 --> 00:07:42,720 Or, the other way to think about it, you could just say-- 145 00:07:42,720 --> 00:07:46,650 So we could just go from this point right here. 146 00:07:46,649 --> 00:07:47,579 We could write this. 147 00:07:47,579 --> 00:07:49,189 We could ignore this, what we did before. 148 00:07:49,189 --> 00:07:53,120 And we could say, this is the same thing as 2 times x to the 149 00:07:53,120 --> 00:07:56,090 eighth to the 1/6 power. 150 00:07:56,089 --> 00:07:58,649 x to the eighth to the 1/6 power. 151 00:07:58,649 --> 00:08:03,569 So this is equal to 2 times x to the-- 8 152 00:08:03,569 --> 00:08:06,709 times 1/6 --8/6 power. 153 00:08:06,709 --> 00:08:09,359 Now we can reduce that fraction. 154 00:08:09,360 --> 00:08:13,780 That's going to be 2 times x to the 4/3 power. 155 00:08:13,779 --> 00:08:17,609 And this and this are completely equivalent. 156 00:08:17,610 --> 00:08:18,660 Why is that? 157 00:08:18,660 --> 00:08:22,530 Because we have 2 times x or 2 times x to the first power 158 00:08:22,529 --> 00:08:24,319 times x to the 1/3 power. 159 00:08:24,319 --> 00:08:28,469 You add 1 to 1/3, you get 4/3. 160 00:08:28,470 --> 00:08:31,350 So hopefully you found this little tutorial on higher 161 00:08:31,350 --> 00:08:32,585 power radicals interesting. 162 00:08:32,585 --> 00:08:35,570 And I think it is useful to kind of see it in prime factor 163 00:08:35,570 --> 00:08:37,640 form and realize, oh, if I'm taking the sixth root, I have 164 00:08:37,639 --> 00:08:40,330 to find a prime factor that shows up at least six times. 165 00:08:40,330 --> 00:08:42,210 And then I could figure out that's 2 to the sixth. 166 00:08:42,210 --> 00:08:45,600 Anyway, hopefully you found this mildly useful.