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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.

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