Dimension subgroups and p-th powers in p-groups

We prove that if the nilpotence class of a p-group is strictly less than p(k) then every product of p(k)-th powers can be written as the p-th power of an element. Scoppola and Shalev have proven the same thing for groups of class strictly less than p(k) - p(k-1). They also provide an example which proves that ours is the best possible result. This is a generalization of the well known fact that in groups of class strictly less than p every product of p-powers is again a p-th power. Along the way we prove results of independent interest on dimension subgroups of p-groups.