Couldn't find answers like that, so I would ask.
At first, let's note that ${\aleph}=c $ in our case. Now, we need to show that: $$(\aleph)^{\aleph_0} < (\aleph_0)^{\aleph} $$
For the left part of this inequality, we know that $\aleph^{\aleph_0}=2^{\aleph_0\aleph_0}=\aleph$.
But for the right part, can we assume that ${\aleph_0}^{\aleph}={\aleph_0}^{\aleph_0}⋅{\aleph_0}^{\aleph}={\aleph}⋅{\aleph_0}^{\aleph}$,
thus making it greater than $\aleph^{\aleph_0}$?
Thanks!