Infinite Cardinal number power.

Question

Let $a$ and $b$ are two infinite cardinal number than can i say that $a^{b}=2^{b}$?

I am thinking so because of there this true for $\aleph_{0}$ and $c=2^{\aleph_{0}}$ as $\aleph_{0}^{c}=2^{c}$ and $c^{\aleph_{0}}=2^{\aleph_{0}}.$

Answer 1

No. Take $a=2^{2^{\aleph_0}}$ and $b=\aleph_0$, then $$a^b=(2^{2^{\aleph_0}})^{\aleph_0}=2^{2^{\aleph_0}\cdot\aleph_0}=2^{2^{\aleph_0}}>2^{\aleph_0}.$$