\((ab)^m\) basically means:
\[ \overbrace{(ab) * (ab) * (ab) * \ldots}^{m} = \overbrace{a * b * a * b * a * b * \ldots}^{m}\]
We can group all the \(a\) together and all the \(b\) together. Each of them appear \(m\) times:
\[ \overbrace{a*a*a*\ldots}^{m} * \overbrace{b*b*b*\ldots}^{m} \]
This proves that \( (ab)^m = a^m * b^m\). This proof only works when \(m\) is an integer.