If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma=\beta^\gamma$.

Question

Can someone suggest a rigorous proof of the following: If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma = \beta^\gamma$.

Answer 1

$$2^\gamma\le\alpha^\gamma\le(2^\gamma)^\gamma=2^{\gamma^2}=2^\gamma$$