Given any infinite cardinal $\alpha$ and finite $n$ we have $\alpha^n = \alpha$.
Also given any cardinal $\alpha$ of the form $\alpha = \gamma ^ \delta$ for $\delta \le \beta$ we have
$$\alpha^\beta = (\gamma ^ \delta)^\beta = \gamma ^{\delta \beta} = \gamma ^{ \beta} = \alpha.$$
I wonder does this hold in general. If say $\alpha > 2^{\beta}$ then do we have $\alpha^\beta = \alpha?$.