  It is well known that a nontrivial commutator in a free group is never a
proper power. We prove a theorem that generalizes this fact and has several
worthwhile corollaries. For example, an equation $[ x_1, y_1] \ldots [ x_k,
y_k] = z^n$, where $n \ge 2k$, in a free product $\mathcal{F}$ of groups
without nontrivial elements of order $\le n$ implies that $z$ is conjugate to
an element of a free factor of $\mathcal{F}$. If a nontrivial commutator in a
free group factors into a product of elements which are conjugate to each other
then all these elements are distinct.
