Simplifying radicals (higher-index roots)
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So far, when we were dealing with radicals we've only been using the square root. We've seen that if I write a radical sign like this and put a 9 under it, this means the principal square root of 9, which is positive 3. Or you could view it as the positive square root of 9. Now, what's implicit when we write it like this is that I'm taking the square root. So I could have also written it like this. I could have also written the radical sign like this and written this index 2 here, which means the square root, the principal square root of 9. Find me something that if I square that something, I get 9. And the radical sign doesn't just have to apply to a square root. You can change the index here and then take an arbitrary root of a number. So for example, if I were to ask you, what-- You could imagine this is called the cube root, or you could call it the third root of 27. What is this? Well, this is some number that if I take it to the third power, I'd get 27. Well, the only number that if you take it to the third power, you get 27 is equal to 3. 3 times 3 times 3 is equal to 27. 9 times 3, 27. So likewise, let me just do one more. So if I have 16-- I'll do it in a different color. If I have 16 and I want to take the fourth root of 16, what number times itself 4 times is equal to 16? And if it doesn't pop out at you immediately, you can actually just do a prime factorization of 16 to figure it out. Let's see. 16 is 2 times 8. 8 is 2 times 4. 4 is 2 times 2. So this is equal to the fourth root of 2 times 2 times 2 times 2. You have these four 2's here. Well, I have four 2's being multiplied, so the fourth root of this must be equal to 2. And you could also view this as kind of the fourth principal root because if these were all negative 2's, it would also work. Just like you have multiple square roots, you have multiple fourth roots. But the radical sign implies the principal root. Now, with that said, we've simplified traditional square roots before. Now we should hopefully be able to simplify radicals with higher power roots. So let's try a couple. Let's say I want to simplify this expression. The fifth root of 96. So like I said before, let's just factor this right here. So 96 is 2 times 48. Which is 2 times 24. Which is 2 times 12. Which is 2 times 6. Which is 2 times 3. So this is equal to the fifth root of 2 times 2 times 2 times 2 times 2. Times 3. Or another way you could view it, is you could view it to a fractional power. You could view it to a fractional power. We've talked about that already. This is the same thing as 2 times 2 times 2 times 2 times 2 times 3 to the 1/5 power. Let me make this clear. Having an nth root of some number is equivalent to taking that number to the 1/n power. These are equivalent statements right here. So if you're taking this to the 1/5 power, this is the same thing as taking 2 times 2 times 2 times 2 times 2 to the 1/5. Times 3 to the 1/5. Now I have something that's multiplied. I have 2 multiplied by itself 5 times. And I'm taking that to the 1/5. Well, the 1/5 power of this is going to be 2. Or the fifth root of this is just going to be 2. So this is going to be a 2 right here. And this is going to be 3 to the 1/5 power. 2 times 3 to the 1/5, which is this simplified about as much as you can simplify it. But if we want to keep in radical form, we could write it as 2 times the fifth root 3 just like that. Let's try another one. Let me put some variables in there. Let's say we wanted to simplify the sixth root of 64 times x to the eighth. So let's do 64 first. 64 is equal to 2 times 32, which is 2 times 16. Which is 2 times 8. Which is 2 times 4. Which is 2 times 2. So we have 1, 2, 3, 4, 5, 6. So it's essentially 2 to the sixth power. So this is equivalent to the sixth root of 2 to the sixth-- that's what 64 is --times x to the eighth power. Now, the sixth root of 2 to the sixth, that's pretty straightforward. So this part right here is just going to be equal to 2. That's going to be 2 times the sixth root of x to the eighth power. And how can we simplify this? Well, x to the eighth power, that's the same thing as x to the sixth power times x squared. You have the same base, you would add the exponents. This is the same thing as x to the eighth. So this is going to be equal to 2 times the sixth root of x to the sixth times x squared. And the sixth root, this part right here, the sixth root of x to the sixth, that's just x. So this is going to be equal to 2 times x times the sixth root of x squared. Now, we can simplify this even more if you really think about. Remember, this expression right here, this is the exact same thing as x squared to the 1/6 power. And if you remember your exponent properties, when you raise something to an exponent, and then raise that to an exponent, that's equivalent to x to the 2 times 1/6 power. Or-- let me write this --2 times 1/6 power, which is the same thing-- Let me not forget to write my 2x there. So I have a 2x there and a 2x there. And this is the same thing as 2x-- it's the same 2x there --times x to the 2/6. Or, if we want to write that in most simple form or lowest common form, you get 2x times x to the-- What do you have here? x to the 1/3. So if you want to write it in radical form, you could write this is equal to 2 times 2x times the third root of x. Or, the other way to think about it, you could just say-- So we could just go from this point right here. We could write this. We could ignore this, what we did before. And we could say, this is the same thing as 2 times x to the eighth to the 1/6 power. x to the eighth to the 1/6 power. So this is equal to 2 times x to the-- 8 times 1/6 --8/6 power. Now we can reduce that fraction. That's going to be 2 times x to the 4/3 power. And this and this are completely equivalent. Why is that? Because we have 2 times x or 2 times x to the first power times x to the 1/3 power. You add 1 to 1/3, you get 4/3. So hopefully you found this little tutorial on higher power radicals interesting. And I think it is useful to kind of see it in prime factor form and realize, oh, if I'm taking the sixth root, I have to find a prime factor that shows up at least six times. And then I could figure out that's 2 to the sixth. Anyway, hopefully you found this mildly useful.