Powers of Disjoint Permutations

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Let $\rho, \sigma$ be disjoint permutations.

Then:
 * $\forall k \in \Z: \paren {\sigma \rho}^k = \sigma^k \rho^k$

Proof
A direct application of Power of Product of Commutative Elements in Group, and the fact that Disjoint Permutations Commute.