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

Options

A. 2
B. 1
C. 0
D. 3

Answer

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

### Step 1: Start with the given equation
We have:
\[ \frac{1}{a + b} = \frac{1}{a} + \frac{1}{b} \]

### Step 2: Find a common denominator on the right side
The common denominator for the right side is \( ab \):
\[ \frac{1}{a + b} = \frac{b + a}{ab} \]
This simplifies to:
\[ \frac{1}{a + b} = \frac{a + b}{ab} \]

### Step 3: Cross-multiply
Cross-multiplying gives us:
\[ 1 \cdot ab = (a + b)(a + b) \]
This simplifies to:
\[ ab = (a + b)^2 \]

### Step 4: Expand the right side
Expanding \( (a + b)^2 \):
\[ ab = a^2 + 2ab + b^2 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 0 = a^2 + b^2 + ab \]

### Step 6: Factor the expression
We can factor this equation:
\[ a^2 + b^2 + ab = 0 \]

### Step 7: Use the identity for \( a^3 - b^3 \)
Recall the identity:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
From our previous step, we know that \( a^2 + ab + b^2 = 0 \).

### Step 8: Substitute into the identity
Substituting this into our identity:
\[ a^3 - b^3 = (a - b)(0) = 0 \]

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