# Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$?

Under what assumptions on an infinite cardinal $\kappa$ we have $$\kappa^\kappa= 2^\kappa?$$

I'm pretty sure this is a repeat, and I know I've posted an answer. In fact, if $\lambda$ satisfies $2\leq \lambda \leq 2^{\kappa}$, then $\lambda^{\kappa}=2^{\kappa}$ by the same argument Asaf uses below. – Arturo Magidin Apr 21 '12 at 22:07
Assuming $\kappa$ is an ordinal the answer is always.
The reason is simple: by Cantor's theorem we have $2<\kappa<2^\kappa$, therefore using exponentiation laws: $$2^\kappa\le\kappa^\kappa\le\left(2^\kappa\right)^\kappa=2^{\kappa\cdot\kappa}=2^\kappa$$