HMMT-Feb 2023 guts, Problem 1

Suppose $a$ and $b$ are positive integers such that $a^{b}=2^{2023}$. Compute the smallest possible value of $b^{a}$.

Solution: Solution: By taking $a=2^{2023}$ and $b=1$, we get $b^{a}=1$, which is clearly the minimum. $\fbox{1}$.

Answer: 1