Regular Expressions: is $a^*a = a^*$ I am practicing simplifying regular expressions. In the list of 12 regular expression identities outlines in this video, I see among them
$AA^*=A^*A$
$ϵ + AA^* = ϵ + A^*A = A^*$
Why do you have to "or" $AA^*$ with the empty string in order for it to equal $A^*$ ? 
Why isnt $AA^* = A^*$ ?
@Wojowu commented, very helpfully, that $AA^* \neq A^*$ because $A^*$ necessarily contains the empty string, while $AA^*$ does not.
However, if A does contain the empty string,ie $A = ϵ + C$, then does $AA^* = A^*$ ?

$a^*$ stands for zero or more repetitions of $a$. You have at least one copy of $a$ in every string in the language $a^*a$, so they are not equivalent.
