How $c^{\aleph_0}=c$

Question

I am reading about coutabaility, uncountability and cardinal numbers. I have attached herewith a screenshot of the wikipedia page.

I am not able to understand how in the third equation we show that $c^{\aleph_0}=c$

I know $c=2^{\aleph_0}$

Can some one please explain it? Rest of the equations are clear to me.

Thanks in advance.

p.s.: Source page : https://en.wikipedia.org/wiki/Cardinality_of_the_continuum

Answer 1

It is only a list of results you can show, and not the actual proof. Once you have $c^{\aleph_0}=c$ the other equalities follows.

To show $c^{\aleph_0}=c$, we use the rule $(\kappa^\lambda)^\mu=\kappa^{\lambda\times\mu}$ and obtain $$c^{\aleph_0}=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\times\aleph_0}=2^{\aleph_0}=c$$