Powers of Disjoint Permutations

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

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

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

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