For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Q$,define $a ^* b = \frac{ab}{2}$.

(N/A) On $Q$,the operation $^*$ is defined by $a ^* b = \frac{ab}{2}$.
Commutativity:
We know that $ab = ba$ for all $a, b \in Q$.
Therefore,$\frac{ab}{2} = \frac{ba}{2}$,which implies $a ^* b = b ^* a$ for all $a, b \in Q$.
Thus,the operation $^*$ is commutative.
Associativity:
For all $a, b, c \in Q$,we have:
$(a ^* b) ^* c = \left(\frac{ab}{2}\right) ^* c = \frac{(\frac{ab}{2})c}{2} = \frac{abc}{4}$.
Also,$a ^* (b ^* c) = a ^* (\frac{bc}{2}) = \frac{a(\frac{bc}{2})}{2} = \frac{abc}{4}$.
Since $(a ^* b) ^* c = a ^* (b ^* c)$ for all $a, b, c \in Q$,the operation $^*$ is associative.