I'd be grateful for any helpful hints, explanations, or explained solutions. Thank you for looking and (hopefully) for replying!!
Since for each real number has exactly one cube and you get the number back by computing the cube root, this is true for all n. This would be different for squares and square roots.
First compute y-x, then the square root, then raise it to the third power, then take the reciprocal.
The expression under the square root is an instance of the formula $\displaystyle a^2 -2ab+b^2 = (a-b)^2$ So you need to write this as a square (of a difference). The square root of a square is ... not quite the term you squared in the first place, but the absolute value of this term: $\displaystyle \left({A^2}\right)^{1/2} = |A|$.
Here's a simpler case: $\displaystyle \left({x^2y^7}\right)^{1/2} = \left({xy^3xy^3y}\right)^{1/2} = |xy^3|\left(y\right)^{1/2} $ Use the same sort of argument.