# Prove or disprove: $L^n=M^n\nRightarrow L=M$ where $L$ and $M$ are languages

In a homework assignment, it's asked

For any alphabet $$\Sigma$$; for all languages $$L$$, $$M$$ on $$\Sigma$$
Prove that $$\forall n>1$$, $$L^n=M^n\nRightarrow L=M$$

The student and I tried in vain to make a proof for $$n=2$$ by exhibiting distinct $$L$$ and $$M$$ with $$L^2=M^2$$; so much that I now think the statement may be wrong. What are your thoughts?

What about $$\Sigma^*$$ and $$\Sigma^*\setminus\{11\}$$?