Periodic factor of hyperbolic groups

It is proved that for any noncyclic hyperbolic torsion-free group $G$ there exists an integer $n(G)$ such that the factor group $G/G^n$ is infinite for any odd $n\geqslant n(G)$. In addition, $\bigcap_{i=1}^\infty G^i=\{1\}$. (Here $G^i$ is the subgroup generated by the $i$th powers of all elements of the groups $G$.)