Is $(a^nb^m)^r$ regular?
The language $L$ is not regular, as can be proved using Nerode's method. Consider the following words $w_n = a^n b$ for $n \in \mathbb{N}$. Then $w_n^2 \in L$, but for $n \neq m$, $w_n w_m \notin L$. Hence any automaton for $L$ must be in a different state after reading each $w_n$, which contradicts its finiteness.