If the sum of ` (a)/( b)` are its reciprocal is 1 and ` a ne 0 , b ne 0` , then the value of ` a^(3) + b^(3)` is

Options

A. 2
B. `-1`
C. 0
D. 1

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 \) given that the sum of \( \frac{a}{b} \) and its reciprocal is equal to 1, with the condition that \( a \neq 0 \) and \( b \neq 0 \).

### Step-by-Step Solution:

1. **Set up the equation**:
We are given that:
\[ \frac{a}{b} + \frac{b}{a} = 1 \]

2. **Combine the fractions**:
To combine the left-hand side, we find a common denominator:
\[ \frac{a^2 + b^2}{ab} = 1 \]

3. **Cross-multiply**:
Cross-multiplying gives us:
\[ a^2 + b^2 = ab \]

4. **Rearrange the equation**:
Rearranging the equation, we have:
\[ a^2 + b^2 - ab = 0 \]

5. **Use the identity for \( a^3 + b^3 \)**:
We can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
From our previous step, we know \( a^2 + b^2 = ab \). Therefore, we can substitute:
\[ a^2 - ab + b^2 = (a^2 + b^2) - ab = ab - ab = 0 \]

6. **Substitute back into the identity**:
Now substituting this back into the identity for \( a^3 + b^3 \):
\[ a^3 + b^3 = (a + b)(0) = 0 \]

### Final Result:
Thus, the value of \( a^3 + b^3 \) is:
\[ \boxed{0} \]