Powers of sets in free groups

We prove that $|A^n|\geqslant c_n\cdot|A|^{[(n+1)/2]}$ for any finite subset $A$ of a free group if $A$ contains at least two noncommuting elements, where the $c_n>0$ are constants not depending on $A$. Simple examples show that the order of these estimates is best possible for each $n>0$.
Bibliography: 5 titles.