According to the fundamental theorem of arithmetic, \(m\) can be represented as:
$$ m = {p_1}^{k_1} * {p_2}^{k_2} * \cdots $$
But we are only interested in the \(p\) in question:
$$ m = p^a * \ldots $$
We can represent \(m^n\) as:
$$\begin{align} m^n &= (p^a * \ldots)^n \\ &= (p^a)^n * (\ldots)^n \\ &= p^{an} * (\ldots)^n \end{align} $$
Therefore, \(p^{an}\) exactly divides \(m^n\).