If * is a binary operation in N defined as a*b`=a^(3)+b^(3)` , then which of the following is true :
(i) * is associative as well as commutative.
(ii) * is commutative but not associative
(iii) * is associative but not commutative
(iv) * is neither associative not commutative.

Answer

To determine whether the binary operation \( * \) defined as \( a * b = a^3 + b^3 \) is associative and/or commutative, we will analyze both properties step by step.

### Step 1: Check for Commutativity
A binary operation \( * \) is commutative if for all \( a, b \in \mathbb{N} \), the following holds:
\[ a * b = b * a \]

**Calculation:**
- Calculate \( a * b \):
\[ a * b = a^3 + b^3 \]
- Calculate \( b * a \):
\[ b * a = b^3 + a^3 \]

Since addition is commutative, we have:
\[ a^3 + b^3 = b^3 + a^3 \]
Thus, \( a * b = b * a \).

**Conclusion for Commutativity:**
The operation \( * \) is commutative.

### Step 2: Check for Associativity
A binary operation \( * \) is associative if for all \( a, b, c \in \mathbb{N} \), the following holds:
\[ (a * b) * c = a * (b * c) \]

**Calculation:**
- Calculate \( (a * b) * c \):
\[ a * b = a^3 + b^3 \]
Now, substituting this into the left side:
\[ (a * b) * c = (a^3 + b^3) * c = (a^3 + b^3)^3 + c^3 \]

- Calculate \( a * (b * c) \):
\[ b * c = b^3 + c^3 \]
Now, substituting this into the right side:
\[ a * (b * c) = a * (b^3 + c^3) = a^3 + (b^3 + c^3)^3 \]

**Conclusion for Associativity:**
Now we need to check if:
\[ (a^3 + b^3)^3 + c^3 \neq a^3 + (b^3 + c^3)^3 \]
These two expressions are not equal in general for arbitrary \( a, b, c \). Thus, \( * \) is not associative.

### Final Conclusion
- The operation \( * \) is commutative but not associative.

### Answer:
The correct option is (ii) \( * \) is commutative but not associative.

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