Simplify the square root of the cube root of 64𝑥 to the 72nd.
We need to work from the inside out. So first, we need to evaluate the cube root. So do we know the cube root of 64? 64 is four times four times four. So the cube root of 64 is four. So in writing our answer, let’s not forget the big square root. So the cube root of 64 we know is four.
So now the cube root of 𝑥 to the 72nd. Now, when finding the cube root of 𝑥 to the 72nd power, we have to be careful. It’s a little different than taking the cube root of 64. When we take the cube root of 64, we need to find a number when multiplied to itself three times we get 64. And that is four times four times four.
Now, the same thing is true with 𝑥 to the 72nd, except when multiplying, we add our exponents. So what number can we add to itself three times that gives us 72? That would be 24. So the cube root of 𝑥 to the 72nd is 𝑥 to the 24th.
So now we need to take the square root of four 𝑥 to the 24th. Now, the square root of four is two. It’s two times two. Now again, 𝑥 to the 24th, what number when added to itself gives us 24? That would be 𝑥 to the 12th because 12 plus 12 is 24. Therefore, our final answer is two 𝑥 to the 12th power.