  In the paper we consider images of finite simple projective special linear
and unitary groups under power words. In particular, we show that if $G\simeq
\PSL_n^\varepsilon (q)$, then for every power words of type $x^M$ there exist
constant $c$ and $N$ such that $\vert \omega(G)\vert >c\frac{\vert G\vert }{n}$
whenether $\vert G\vert >N$.
