Why the domain of the cube root function are all the real numbers?

Question

since it can also be written as x^(1/3) and therefore 1/(x^3) and this would not make sense for x=0 because of the division with 0. So why is 0 in the domain?

Answer 1

$y=x^\frac13\implies x=y^3$, on the contrary, $y=\frac{1}{x^3}\implies x=(\frac 1y)^\frac13$

They aren't the same (except when $x=1$), because $\frac{1}{a^n}$ is written as $a^{-n}$, not $a^{\frac1n}$

Notice that $\frac{a^m}{a^n}=a^ma^{-n}=a^{m-n}$ holds with this notation