For each binary operation `'**'` defined below, determine whether `'**'` is commutative and whether `'**'` is associative :
(v) On `Z^(+)`, define `'**'` by `a**b=2^(ab)`

Answer

To determine whether the binary operation \( ** \) defined by \( a ** b = 2^{(ab)} \) on \( \mathbb{Z}^+ \) is commutative and associative, we will analyze each property step by step.

### Step 1: Check for Commutativity

**Definition of Commutativity**: A binary operation \( ** \) is commutative if for all \( a, b \in \mathbb{Z}^+ \), \( a ** b = b ** a \).

**Calculation**:
1. Compute \( a ** b \):
\[ a ** b = 2^{(ab)} \]
2. Compute \( b ** a \):
\[ b ** a = 2^{(ba)} \]
3. Since multiplication is commutative (i.e., \( ab = ba \)), we have:
\[ a ** b = 2^{(ab)} = 2^{(ba)} = b ** a \]

**Conclusion for Commutativity**: Since \( a ** b = b ** a \) for all \( a, b \in \mathbb{Z}^+ \), the operation \( ** \) is commutative.

### Step 2: Check for Associativity

**Definition of Associativity**: A binary operation \( ** \) is associative if for all \( a, b, c \in \mathbb{Z}^+ \), \( (a ** b) ** c = a ** (b ** c) \).

**Calculation**:
1. Compute \( (a ** b) ** c \):
- First calculate \( a ** b \):
\[ a ** b = 2^{(ab)} \]
- Now calculate \( (a ** b) ** c \):
\[ (a ** b) ** c = 2^{(ab)} ** c = 2^{(2^{(ab)} \cdot c)} \]

2. Compute \( a ** (b ** c) \):
- First calculate \( b ** c \):
\[ b ** c = 2^{(bc)} \]
- Now calculate \( a ** (b ** c) \):
\[ a ** (b ** c) = a ** (2^{(bc)}) = 2^{(a \cdot 2^{(bc)})} \]

3. Compare \( (a ** b) ** c \) and \( a ** (b ** c) \):
- We have:
\[ (a ** b) ** c = 2^{(2^{(ab)} \cdot c)} \]
\[ a ** (b ** c) = 2^{(a \cdot 2^{(bc)})} \]
- These two expressions are not equal in general because the exponentiation behaves differently.

**Conclusion for Associativity**: Since \( (a ** b) ** c \neq a ** (b ** c) \) for all \( a, b, c \in \mathbb{Z}^+ \), the operation \( ** \) is not associative.

### Final Conclusion
- The operation \( a ** b = 2^{(ab)} \) is **commutative** but **not associative**.

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