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I am trying to prove the following statement:

If $S$ and $T$ are any regular expressions over a 1-letter alphabet and if $n$ is a natural, then the languages $(ST)^n$ and $S^nT^n$ are equal.

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Hint: Show that any two regexps over a $1$-letter alphabet commute, i.e. $PQ = QP$ for every two regexps $P,Q$.

Hint to the hint: Show that any two words over a $1$-letter alphabet commute, i.e. $ab = ba$ for every two words $a,b$.

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