If `(1)/( a) - (1)/( b) = (1)/( a - b)` then the value of ` a^(3) + b^(3) `is

Options

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

Answer

To solve the equation \( \frac{1}{a} - \frac{1}{b} = \frac{1}{a - b} \) and find the value of \( a^3 + b^3 \), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ \frac{1}{a} - \frac{1}{b} = \frac{1}{a - b} \]

### Step 2: Find a common denominator
The left-hand side can be rewritten using a common denominator:
\[ \frac{b - a}{ab} = \frac{1}{a - b} \]

### Step 3: Cross-multiply
Cross-multiplying gives:
\[ (b - a)(a - b) = ab \]

### Step 4: Simplify the left-hand side
Notice that \( (b - a)(a - b) = -(a - b)^2 \):
\[ -(a - b)^2 = ab \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ (a - b)^2 + ab = 0 \]

### Step 6: Analyze the equation
Since \( (a - b)^2 \) is always non-negative, the only way for the sum to equal zero is if both terms are zero:
\[ (a - b)^2 = 0 \quad \text{and} \quad ab = 0 \]
This implies \( a = b \) and either \( a = 0 \) or \( b = 0 \).

### Step 7: Calculate \( a^3 + b^3 \)
Using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):
- Since \( a = b \), we can substitute \( b \) with \( a \):
\[ a^3 + b^3 = 2a^3 \]
- If \( a = 0 \), then:
\[ a^3 + b^3 = 0^3 + 0^3 = 0 \]

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