Is the language $(a^n)^mb^n$ context free? $(a^n)^mb^n$  for $m,n\ge 1$
This can be rewritten as $a^{nm}b^n   $
i.e. number of $a$'s is a multiple of number of $b$'s, or for every m $a$'s there is one $b$. I thnk this language can be accepted by NPDA. For every m $a$'s one a can be pushed into the stack, and similarly one can be popped on encountering a $b$. But the NPDA needs to try this for every possible m, where m varies from 1 to maximum string length of $a$'s.
The difficulty of trying every possible $m$ is, in fact, insurmountable. That language is not context-free.

For the sake of contradiction, assume it is context-free. Let $p$ be its pumping length. Consider the word $a^{p^2p}b^{p^2}$. I will let you take it from here.