I'm not sure I've got this right. When proving $a^n \mid b^n \Rightarrow a \mid b$, can we do this indirectly? In short,

"Suppose $a$ does not divide $b$, this implies that $a^n$ does not divide $b^n$. But $a^n \mid b^n$, hence $a$ divides $b$."

How about $n^n \mid m^m \Rightarrow n \mid m$? Can we do this the same way?

This is false: $4^4$ divides $10^{10}$ but $4$ does not divide $10$.