Let $F=<X>$ be a free group and $v, w \in F$ such that $v^n \cdot w^m = w^m \cdot v^n$. Show that $v\cdot w = w\cdot v$.
I didn't think it'd be very hard, but it turns out I'm stuck. I've thought of using induction on the lengths of $v$ and $w$, or induction on $n$ and $m$ but I didn't manage to get anything out of it. So if the solution includes induction and you don't want to give it out immediately but with a hint, please be a little more specific than just suggesting to use induction. Thanks in advance for your help.