Power of two commuting elements in a group is the binary operation of each of the two elements raised to that power

Question

Let $(G,\ast)$ be a group and let $n\in\aleph$. Prove that if g, h $\in G$ commute, then $(g\ast h)^n$=$g^n\ast h^n$

Answer 1

I'll omit the $*$ symbol, add it if you prefer.

First prove that $gh^n=h^ng$ by induction on $n$.

Then observe that $$ (gh)^{n+1}=(gh)^n(gh)\overset{\text{IH}}{=}(g^nh^n)(gh)=g^n(h^ng)h=\dots $$ where $\overset{\text{IH}}{=}$ denotes application of the induction hypothesis.