Power of Product in Abelian Group

Theorem
Let $\struct {G, \circ}$ be an abelian group.

Then:
 * $\forall x, y \in G: \forall k \in \Z: \paren {x \circ y}^k = x^k \circ y^k$

Additive Notation
This can also be written in additive notation as:

Proof
By definition of abelian group, $x$ and $y$ commute.

That is:
 * $x \circ y = y \circ x$

The result follows from Power of Product of Commutative Elements in Group.