Certain Normal Subgroups of a Free Group

Certain Normal Subgroups of a Free Group

Let $F$ be a free group on some set $X$. For any $x \in F$ and $n \in
\mathbb{Z}$, we have that $\langle x^n \rangle \lhd F$. Why does this
hold?
For any word $w \in F$, we have $wx^n w^{-1} = (wxw^{-1})^n$ and hence it
seems to suffice to show that $wxw^{-1} = x^k$ for some $k \in
\mathbb{Z}$, but I am not sure why this should be the case.