Let's say I have $4$ sets: $A,B,C,D$, is there a way to determine whether or not $\left|A^B\right|>\left|C^D\right|$?
I thought about this question because it is easy to show that $\left|n^\Bbb N\right|=\left|m^\Bbb N\right|$ forall $n,m\in\Bbb N$, but also $\left|\Bbb N^\Bbb N\right|$ is equal to the above. I was also told that $\left|\Bbb R^\Bbb N\right|$ is equal to them(although I didn't prove that one), so $\left|n^\Bbb N\right|=\left|\Bbb N^\Bbb N\right|=\left|\Bbb R^1\right|=\left|\Bbb R^\Bbb N\right|$, it means that simply compare $|B|$ with $|D|$ and $|A|$ with $|C|$ won't be enough.
Also, what about the other direction? If $\left|A^B\right|>\left|C^D\right|$ what can we say about the relation of the cardinality of $A,B,C,D$?