Powers of Commutative Elements in Groups

Theorem

Let $\left ({G, \circ}\right)$ be a group.

Let $a, b \in G$ such that $a$ and $b$ commute.

Then the following results hold:

Commutativity of Powers

$\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$

Product of Commutative Elements

$\forall n \in \Z: \left({a \circ b}\right)^n = a^n \circ b^n$