> [!NOTE] Theorem (Power & Product rules) > Let $G$ be a [[Groups|group]]. Let $a\in G.$ If $m, n\in\mathbb{Z},$ then $(a^{m})^{n}=a^{mn} $ where $a^{n}$ denotes the $n$th [[Integer Power of Group Element|power]] of $a.$ **Proof**: Consider the different possibilities of the signs of $m$ and $n.$