Powers of Disjoint Permutations

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

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

Then:
 * $\forall k \in \Z: \left({\sigma \rho}\right)^k = \sigma^k \rho^k$

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