Cyclotomic Polynomial of Index times Prime Power

Theorem
Let $n,k\geq1$ be natural numbers.

Let $p$ be a prime number.

Let $\Phi_n$ denote the $n$th cyclotomic polynomial.

Then $\Phi_{p^kn}(x) = \begin{cases} \Phi_n\left( x^{p^k} \right)&\text{if }p\mid n\\ \dfrac{\Phi_n\left( x^{p^k} \right)}{\Phi_n\left( x^{p^{k-1}} \right)}&\text{if }p\nmid n\end{cases}$