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In the paper, we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that, ifG  ≃    PSL  n  ε   ⁡   (  q  ){G\simeq\operatorname{PSL}_{n}^{\varepsilon}(q)}   , then, for every power word of typex  M{x^{M}}   , there exist constants  c  and  N  such that|   ω  ⁢   (  G  )    |   &gt;   c  ⁢ln  ⁡   (  n  )    ⁢   |  G  |    n{\lvert\omega(G)\rvert&gt;c\frac{\ln(n)\lvert G\rvert}{n}}    whenever|  G  |   &gt;  N{\lvert G\rvert&gt;N}   .

On Shalev's conjecture for type 𝐴 𝑛 and 2 𝐴 𝑛