Are the functions $f(x)=x^\frac{m}{n}$ and $g(x)= \sqrt[n]{x^m}$ equal for $m \in \mathbb{Z}$ and for $n \in \mathbb{N}^*$ operating for $x \in \mathbb{R}$? Some argue that they are not since their domain can be different. However, my argument is that since one form can be writtern as the other same restrictions can be applied. If the proposes logic is followed this means that:
$\sqrt[n]{x^m} \Rightarrow x^\frac{m}{n}$ holds only and the following does not
$\sqrt[n]{x^m} \iff x^\frac{m}{n}$. I'd like to hear some opinions