Since \(\gcd(a, b) = 1\), then \(a\) and \(b\) don't have any common prime factors:
$$\begin{align} a &= p_1^{a_1} * p_3^{a_3} * p_4^{a_4} \\ b &= p_2^{b_2} * p_5^{b_5} \end{align}$$
If we raise both integers to some integer power, there would still be no common prime factors:
$$\begin{align} a^m &= p_1^{ma_1} * p_3^{ma_3} * p_4^{ma_4} \\ b^n &= p_2^{nb_2} * p_5^{nb_5} \end{align}$$
This proves that \(\gcd(a^m, b^n) = 1\).