Let $G$ be a torsion-free nonelementary Gromov hyperbolic group and let $X$ be a finite set. For any nontrivial element $w$ of the free product $G \ast F(X)$, there exists an element $g$ of $G^X$ such that $w(g)$ is nontrivial in $G$. We investigate the problem of estimating the minimal word-length of such an element $g$, in terms of the word-length of $w$.