Modular congruence rules

Question

May take exponent of both sides of a modular congruence? For instance, may I write

$$n^2\equiv-1\mod p \quad \rm as \quad(n^2)^{2k+1}\equiv (-1)^{2k+1}\mod p ?$$

Answer 1

If $p|(a-b)$ does $p|a^k-b^k$?

Answer 2

Sure, if $a\equiv b \mod n$ then for every polynomial $f\in {\mathbb{Z}}[x]$ you have $f(a)\equiv f(b) \mod n .$