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?