Find the value of the cube root of 47 cubed.
So notice, on the inside of the parentheses, we’ve a cube root. And on the outside of the parentheses, we’re cubing it, raising it to the third power. And what will happen is those will cancel each other, because those are inverse operations, making our final answer 47. And here’s why.
With inverse operations, they essentially cancel each other out. So if we cube something that has been cube-rooted, we get itself, so we would get 𝑎. And even if we did the reverse, if we took the cube root of something cubed, it’d still be itself, just 𝑎.
And we can show why by rewriting the cube root function. If we cube-root a number, it’s actually taking that number and raising it to the one-third power. So if we replace the cube root of 𝑎 with 𝑎 to the one-third power, we can now see that we need to multiply one-third and three. And one-third times three is the same thing as one-third times three over one. And we get three over three, which is simply equal to one. So we end up with 𝑎 to the first power, which is 𝑎 just like we had said.
So now let’s look at the second way, the same thing but in reverse. So we will rewrite the cube root as 𝑎 one-third power. So we will be multiplying three times one-third, which still results in 𝑎 to the first power, which is 𝑎. So we were asked to find the value of the cube root of 47 all cubed. We would just get 47 because the cube root and cubing something are inverse operations.