Exponentiation is distributive over multiplication, but it isn't commutative or associative like addition and multiplication are.
Is there a binary operation that is distributive over multiplication, and also commutative and/or associative?
In order to find one such operation, I assumed that there is an identity element. An easier question than the one above is: Is the identity element $0$, $1$, or neither?
If you find an operation that works, can you then find a commutative and/or associative operation that is distributive over that?