Is $\alpha^{\beta}=2^{\beta}$ for all infinite cardinalities $\alpha, \beta$?

Question

Is $\alpha^{\beta}=2^{\beta}$ for all infinite cardinalities $\alpha, \beta$?

I was wondering about this since I've encountered examples of times where this holds, but I can't seem to prove it myself and I'm pretty skeptical that it's true in general.

Can anyone shed light on this please? What are the limitations of this?

Thanks!

Answer 1

Not at all. For example let $\alpha=2^{2^\beta}$ then we have: $$\alpha^\beta=\left(2^{2^\beta}\right)^\beta=2^{2^\beta\cdot\beta}=2^{2^\beta}=\alpha.$$

If $2<\alpha\leq 2^\beta$ then $2^\beta\leq\alpha^\beta\leq2^\beta$ and we have equality. Otherwise $2^\beta<\alpha$ and $(2^\beta)^\beta=2^\beta<\alpha\leq\alpha^\beta$.