Question: Cube Root of a Decimal

I converted 0.000008 into the fraction 8/1,000,000 because I can easily determine the individual cube roots of 8 and 1,000,000.

Also, when I scanned the answer choices (ALWAYS scan the answer choices before performing any calculations!), I see that the answers are written as fractions. So, that was a hint to convert the given decimal to a fraction.

That said, your approach of first determining the cube root of 0.000008 and then dealing with the exponent (-1) is a perfectly valid approach.

So, we get: [cuberoot(0.000008)]^(-1) = [0.02]^(-1)
= [2/100]^(-1)
= [1/50]^(-1)
= 50

That approach will work also.
cuberoot(0.000008) = cuberoot(8 X 10^-6)
= cuberoot(8) x cuberoot(10^-6)
= 2 x 10^-2
= 2 x 1/100
= 2/100
= 1/50

So, [cuberoot(0.000008)]^-1 = (1/50)^-1 = 50

There are a few ways to evaluate cuberoot(10^-6)

We might ask, "What expression times itself THREE TIMES will yield the result of 10^-6?"
Aside: This technique is no different from evaluating something like cuberoot(8). We ask, "What number times itself THREE TIMES will yield a result of 8?"
In other words, (?)(?)(?) = 8
Since (2)(2)(2) = 8, we know that cuberoot(8) = 2

So, let's go back to cuberoot(10^-6)
We can write: (?)(?)(?) = 10^-6
Well, (10^-2)(10^-2)(10^-2) = 10^-6
So, cuberoot(10^-6) = 10^-2

Another approach: recognize that cuberoot(N) = N^(1/3)
So, cuberoot(10^-6) = (10^-6)^(1/3)
Apply the Power of a Power rule, we get: (10^-6)^(1/3) = 10^(-6 x 1/3)
= 10^(-6/3)
= 10^-2
= 1/10