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

Please delete this question. I know the answer.

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    Once posting a question, it might be useful for future visitors. If you know the answer it is fine to post an answer on your own.2012-04-21
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    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.2012-04-21

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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: $$2^\kappa\le\kappa^\kappa\le\left(2^\kappa\right)^\kappa=2^{\kappa\cdot\kappa}=2^\kappa$$