In "Naive Set Theory" Halmos gives an exercise:
Prove that if $a, b$ are cardinal numbers, $a$ is infinite, $b$ is finite, then $a^b = a$
I struggle to use the known properties of cardinal numbers as well as induction on $b$ (if it is applicable here at all) or thinking about it in terms of sets of functions.
It is written in the Wikipedia that this property requires the Axiom of Choice, and in the book it is assumed.