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

Options

A. `-2`
B. 0
C. 2
D. 4

Answer

To solve the equation \(\frac{1}{x+y} = \frac{1}{x} + \frac{1}{y}\), we can follow these steps:

### Step 1: Rewrite the equation
Starting with the given equation:
\[ \frac{1}{x+y} = \frac{1}{x} + \frac{1}{y} \]

### Step 2: Find a common denominator for the right side
The common denominator for \(\frac{1}{x} + \frac{1}{y}\) is \(xy\):
\[ \frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy} = \frac{x+y}{xy} \]

### Step 3: Set the two sides equal
Now we can set the two sides equal:
\[ \frac{1}{x+y} = \frac{x+y}{xy} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ 1 \cdot xy = (x+y)(x+y) \]
which simplifies to:
\[ xy = (x+y)^2 \]

### Step 5: Expand the right side
Expanding \((x+y)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\):
\[ xy = x^2 + 2xy + y^2 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 0 = x^2 + y^2 + xy \]

### Step 7: Use the identity for \(x^3 - y^3\)
Recall the identity:
\[ x^3 - y^3 = (x-y)(x^2 + xy + y^2) \]
Since we have \(x^2 + y^2 + xy = 0\), we can substitute this into the identity:
\[ x^3 - y^3 = (x-y)(0) = 0 \]

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