Is $(a^nb^m)^r$ regular?

Problem Detail: 

I took my theory of computation exams a few weeks ago, and this was one of the questions:

Assume language $L=\{(a^nb^m)^r \mid n,m,r\ge 0\}$

Is L regular? If yes provide a regular expression or an automaton for it.

After I briefly asked him the answer after the exam, it appears it really is regular (I believe he said the expression is the simple $(a^*b^*)^*$). However I cannot seem to understand why that is. The way I see it, its concatenating $a^nb^m$ r times, like this:

$a^nb^ma^nb^ma^nb^m...a^nb^ma^nb^m$,

which isn't regular since there is no way for an automaton to recall n and m every time. Where am I at fault here?

EDIT: I talked to the professor again, he admitted it was a mistake. The language is indeed not regular.

Asked By : Exci

Answered By : Yuval Filmus

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.

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Question Source : http://cs.stackexchange.com/questions/4837