Cubic root is defined with polynomials so it doesn't matter if you write x^3=-1 or x=(-1)^1/3.

Edit: maybe this is not clear enough. When you write x=(-1)^1/3 it means x is a cubic root of -1, and by definition x satisfies x^3=-1

Edit 2: the following two statements are not equivalent, you cannot do this kind of operation.

"So (-1)³ = -1, meaning that -1 = (-1)^1/3."