Verbal subgroups of hyperbolic groups have infinite width

Let G be a non-elementary hyperbolic group. Let w be a proper group word. We show that the width of the verbal subgroup w(G) = <w[G]> is infinite. That is, there is no l ε Z such that any g ε w(G) can be represented as a product of at most l values of w and their inverses. As a consequence, we obtain the same result for a wide class of relatively hyperbolic groups.