Let $F$ be a free group. Prove that the subgroup generated by all nth powers, $\{x^n|x\in F \}$ is a normal subgroup of $F$.

Question

What I feel confused is that:

For example, if $\{x^n|x\in F \}$ is normal, then the reduced word $yx^ny^{-1}\in \{x^n|x\in F \}$, How could it be possible?

This is also the exercise 3 on page 68 of Hungerford's textbook Algebra.

Answer 1

Hint: $$\left(yxy^{-1}\right)^{n}=yx^{n}y^{-1}$$