For each binary operation `'**'` defined below, determine whether `'**'` is commutative and whether `'**'` is associative :
(vi) On `Z^(+)`, define `'**'` by `a**b=a^(b)`

Answer

To determine whether the binary operation '**' defined by \( a ** b = a^b \) on \( \mathbb{Z}^+ \) is commutative and associative, we will analyze each property step by step.

### Step 1: Check for Commutativity
A binary operation is commutative if:
\[ a ** b = b ** a \]
for all \( a, b \in \mathbb{Z}^+ \).

Using the definition of the operation:
- \( a ** b = a^b \)
- \( b ** a = b^a \)

Now we need to check if \( a^b = b^a \).

**Example:**
Let’s take \( a = 2 \) and \( b = 3 \):
- \( a ** b = 2^3 = 8 \)
- \( b ** a = 3^2 = 9 \)

Since \( 8 \neq 9 \), we conclude that:
\[ a ** b \neq b ** a \]
Thus, the operation '**' is **not commutative**.

### Step 2: Check for Associativity
A binary operation is associative if:
\[ (a ** b) ** c = a ** (b ** c) \]
for all \( a, b, c \in \mathbb{Z}^+ \).

Using the definition of the operation:
- Left-hand side: \( (a ** b) ** c = (a^b) ** c = (a^b)^c \)
- Right-hand side: \( a ** (b ** c) = a ** (b^c) = a^{(b^c)} \)

Now we need to check if \( (a^b)^c = a^{(b^c)} \).

Using the laws of exponents:
- The left-hand side simplifies to \( a^{bc} \) (since \( (a^b)^c = a^{bc} \)).
- The right-hand side is \( a^{(b^c)} \).

Now, we need to determine if \( a^{bc} = a^{(b^c)} \) for all \( a, b, c \in \mathbb{Z}^+ \).

**Example:**
Let’s take \( a = 2 \), \( b = 2 \), and \( c = 3 \):
- Left-hand side: \( (2 ** 2) ** 3 = (2^2) ** 3 = 4 ** 3 = 4^3 = 64 \)
- Right-hand side: \( 2 ** (2 ** 3) = 2 ** (2^3) = 2 ** 8 = 2^8 = 256 \)

Since \( 64 \neq 256 \), we conclude that:
\[ (a ** b) ** c \neq a ** (b ** c) \]
Thus, the operation '**' is **not associative**.

### Conclusion
The binary operation defined by \( a ** b = a^b \) is neither commutative nor associative.