Why does CubeRoot and power of 1/3 give different answers in Mathematica?

Question

I have these 2 functions, which should give identical answers:

GPrime[x_] := (1/CubeRoot[x]) + 1

GPrime2[x_] := (1/(x^(1/3))) + 1

However, given this:

{GPrime[-8],GPrime2[-8]}

i get 2 different answers:

$$\frac{1}{2} , 1 - \frac{1}{2}(-1)^{2/3}$$

why is the answer not $\frac{1}{2}$ for both functions?

Answer 1

For complex numbers $z^w$ is multivaluated. The principal value of $z^w$ is given by $e^{w \log(z)}$ where the principal branch of $\log$ is taken.

Since Mathematica works with complex numbers by default, then $(-2)^{1/3} = e^{\tfrac{1}{3}\log(-2)} = e^{\tfrac{1}{3}(\log(2) + \pi i)}=2^{1/3}(\frac{1+i\sqrt{3}}{2})$.

But when the base $z$ is negative and the exponent is of the form $w=\frac{1}{2n+1}$ one could prefer to use the real $(2n+1)$-th root of $z$. Mathematica have the convenient function CubeRoot for the case of cube root and Surd for a more general case.

Mathematica's choice could look weird, but that choice allows the function $z^w$ be treated in a general nice way. For example, for fixed $z$ is a continuous function of $w$, while taking the real roots won't even make sense for most exponents.