How to prove that concatenating a language A and A* is commutative?
$AA^*$ is the set of all strings consisting of a string in A, followed by zero or more strings in A, which means all strings consisting of one or more strings in A. $A^*A$ is the set of all strings consisting of zero or more strings in A, followed by one string in A, which also means all strings consisting of one or more strings in A. So both are the same.