is ${x^{2/3}}$ the same as ${\sqrt[3]{(x^2)}}$ or ${(\sqrt[3]x)^2}$? domain?

Question

is ${x^{2/3}}$ the same as ${\sqrt[3]{x^2}}$ and ${(\sqrt[3]x)^2}$?
and what is domain of $x$?

Wolfram Alpha shows different results for ${x^{2/3}}$ and ${\sqrt[3]{x^2}}$ representations.

Is domain dependant just on representation!?

Answer 1

It would appear that Wolfram Alpha is secretly considering the functions in different contexts; for $x^{2/3}$ it is treating the $x$ as a complex variable, which can cause all kinds of shenanigans, whereas $\sqrt[3]{x^2}$ has $x$ as a purely real variable, since there is no way that $x^2$ can be negative. Thus in the first case you must consider cube roots of $-1$, and in the second case you may always take the positive real root.

In short, the domain is NOT dependent on representation. Wolfram Alpha just likes to cut corners.