I am stuck on the following problem that says:
Assuming the Generalized Continuum Hypothesis (GCH), that is, the statement $2^{\aleph_{\alpha}}$ = $\aleph_{\alpha+1}$ for every ordinal $\alpha$, find the corresponding $\aleph$ numbers of the following computations in cardinal arithmetic:
$a$) $\aleph_0^{\aleph_1^{\aleph_2}}$
$b)$ $(\aleph_{\aleph_0})^{\aleph_{\aleph_1}}$
$c)$ $(\aleph_{\aleph_\omega})^{2^{\aleph_7}}$
$d$) $(\aleph_{\aleph_1})^{\aleph_2}$
My Attempt:
For parts $a)$ and $b)$ I know that
If GCH holds, then if $x,k$ are both infinite cardinals:
1) if $k\leq x$, then $k^x=x+$
2) if $cfk\leq x$, then $k^x=k+$
3) if $x<cfk$, then $k^x=k$
So, I found that, $a)=\aleph_4$, $b)$ $\aleph(\aleph_{\aleph_1})$
But, in parts $c)$ and $d)$ I couldn't compute the cofinity of these numbers,
Can someone help me out? Thanks in advance for your time!