Let $\alpha$ be an irrational number. Denote by $\mu(\alpha)$ its irrationality measure. Can one say anything about $\mu(\alpha^n)$ for every $n\in\mathbb N$?

Even more, one knows that $\mu(e)=2$. Can one say anything for $\mu(e^{p/q})$ for $\frac pq\in\mathbb Q^*$?