TITLE: Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$?
QUESTION [10 upvotes]: Under what assumptions on an infinite cardinal $\kappa$ we have
$$\kappa^\kappa= 2^\kappa?$$
Please delete this question. I know the answer.

REPLY [16 votes]: Assuming $\kappa$ is an ordinal the answer is always.
The reason is simple: by Cantor's theorem we have $2<\kappa&lt2^\kappa$, therefore using exponentiation laws: $$2^\kappa\le\kappa^\kappa\le\left(2^\kappa\right)^\kappa=2^{\kappa\cdot\kappa}=2^\kappa$$