Consider a binary operation `'**'` on N defined as : `a**b=a^(3)+b^(3)`. Then :

Options

A. is `'**'` both associative and commutative?
B. is `'**'` commutative but not associative?
C. is `'**'` associative but not commutative?
D. Is `'**'` neither commutative nor associative?

Answer

To solve the problem, we need to analyze the binary operation defined as \( a ** b = a^3 + b^3 \) on the set of natural numbers \( \mathbb{N} \). We will check if this operation is commutative and associative.

### Step 1: Check for Commutativity
To check if the operation is commutative, we need to verify if \( a ** b = b ** a \) for all \( a, b \in \mathbb{N} \).

1. Calculate \( a ** b \):
\[ a ** b = a^3 + b^3 \]

2. Calculate \( b ** a \):
\[ b ** a = b^3 + a^3 \]

3. Compare \( a ** b \) and \( b ** a \):
\[ a ** b = a^3 + b^3 = b^3 + a^3 = b ** a \]

Since \( a ** b = b ** a \), the operation is **commutative**.

### Step 2: Check for Associativity
To check if the operation is associative, we need to verify if \( (a ** b) ** c = a ** (b ** c) \) for all \( a, b, c \in \mathbb{N} \).

1. Calculate \( (a ** b) ** c \):
- First, find \( a ** b \):
\[ a ** b = a^3 + b^3 \]
- Now, apply the operation with \( c \):
\[ (a ** b) ** c = (a^3 + b^3) ** c = (a^3 + b^3)^3 + c^3 \]

2. Calculate \( b ** c \):
\[ b ** c = b^3 + c^3 \]
- Now, apply the operation with \( a \):
\[ a ** (b ** c) = a ** (b^3 + c^3) = a^3 + (b^3 + c^3)^3 \]

3. Compare \( (a ** b) ** c \) and \( a ** (b ** c) \):
- We have:
\[ (a ** b) ** c = (a^3 + b^3)^3 + c^3 \]
\[ a ** (b ** c) = a^3 + (b^3 + c^3)^3 \]

These two expressions are not equal in general, which means:
\[ (a^3 + b^3)^3 + c^3 \neq a^3 + (b^3 + c^3)^3 \]

Thus, the operation is **not associative**.

### Conclusion
The binary operation \( ** \) defined as \( a ** b = a^3 + b^3 \) is **commutative** but **not associative**.