This is false for torsion free uniform lattices in rank one semi simple Lie groups and large n by a result of Olshanskii, which implies that the normal subgroup generated by nth powers is infinite index for large n.

This is false for uniform lattices in rank one semi simple Lie groups and large $n$ by a result of Ivanov and Olshanskii, which implies that the normal subgroup generated by $n$th powers is infinite index for certain large $n$.