Suppose $F_n$ is a free group of rank $n$, and $a$ is an element of $F_n$, such that $\exists b \in F_n(b^p=a)$ and $\exists c \in F_n(c^q=a)$, where $p$ and $q$ are coprime integers. Does there always $\exists d \in F_n(d^{pq}=a)$?
It seems to be so, but I failed to find any correct proof of this statement.
Any help will be appreciated.
Consider the subgroup $\langle b,c \rangle$ of $F$. As a subgroup of a free group, it is itself free, but $a$ is in its centre, and the only free group with nontrivial centre is the infinite cyclic group.
So $\langle b,c \rangle = \langle g \rangle$ for some $g \in F$, and $a = g^k$ for some $k \in {\mathbb Z}$. Since $a$ is both a $p$-th power and a $q$-th power, $p$ and $q$ both divide $k$, and so $a$ is also a $(pq)$-th power.