For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Z^+$,define $a ^* b = a^b$.

(NONE) On $Z^+$,the operation $^*$ is defined by $a ^* b = a^b$.
$1$. Commutativity:
We check if $a ^* b = b ^* a$ for all $a, b \in Z^+$.
Consider $a = 1$ and $b = 2$.
$1 ^* 2 = 1^2 = 1$
$2 ^* 1 = 2^1 = 2$
Since $1 \neq 2$,$1 ^* 2 \neq 2 ^* 1$.
Therefore,the operation $^*$ is not commutative.
$2$. Associativity:
We check if $(a ^* b) ^* c = a ^* (b ^* c)$ for all $a, b, c \in Z^+$.
Consider $a = 2, b = 3, c = 4$.
$(2 ^* 3) ^* 4 = (2^3) ^* 4 = 8 ^* 4 = 8^4 = (2^3)^4 = 2^{12}$.
$2 ^* (3 ^* 4) = 2 ^* (3^4) = 2 ^* 81 = 2^{81}$.
Since $2^{12} \neq 2^{81}$,$(2 ^* 3) ^* 4 \neq 2 ^* (3 ^* 4)$.
Therefore,the operation $^*$ is not associative.