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# Radicals and exponents

Sal explains the relationship between radical notation (like ∜7) and unit-fraction exponents (like 7^¼). Created by Sal Khan and CK-12 Foundation.

Video transcript

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.