For example Prove by induction that the operation of raising to the power m$\in$ $\mathbb{N}$ is well defined in $\mathbb{Z}_n$
$\forall$m$\in$ $\mathbb{N}$ $\forall$[x]$\in$ $\mathbb{Z}$/$_{{\sim}n}$
so that we have $[x^m]$=$[x]^m$
I am completely at lost here with this proof can someone please help.
If I understand your question correctly, suppose $p = x^m \bmod n$. Then show that $(x+n)^m \bmod n = p$. With your basis case at $x \in \mathbb{Z}_n$, this should establish your proposition.