If r and s are variables satisfying the equation `(1)/(r+s) =1/r+1/s. ` The value of `((r )/(s ))^(3)` is equal to :

Options

A. 1
B. `-1`
C. 3
D. not possible to determine

Answer

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

### Step 1: Rewrite the right-hand side
Start by rewriting the right-hand side of the equation:
\[ \frac{1}{r} + \frac{1}{s} = \frac{s + r}{rs} \]
So the equation becomes:
\[ \frac{1}{r+s} = \frac{r+s}{rs} \]

### Step 2: Cross-multiply
Cross-multiply to eliminate the fractions:
\[ 1 \cdot (rs) = (r+s)(r+s) \]
This simplifies to:
\[ rs = (r+s)^2 \]

### Step 3: Expand the right-hand side
Now, expand the right-hand side:
\[ rs = r^2 + 2rs + s^2 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 0 = r^2 + s^2 + rs \]

### Step 5: Divide by \(rs\)
Now, divide the entire equation by \(rs\) (assuming \(rs \neq 0\)):
\[ 0 = \frac{r^2}{rs} + \frac{s^2}{rs} + 1 \]
This simplifies to:
\[ 0 = \frac{r}{s} + \frac{s}{r} + 1 \]

### Step 6: Let \(x = \frac{r}{s}\)
Let \(x = \frac{r}{s}\). Then the equation becomes:
\[ 0 = x + \frac{1}{x} + 1 \]

### Step 7: Multiply through by \(x\)
Multiply through by \(x\) to eliminate the fraction:
\[ 0 = x^2 + 1 + x \]
Rearranging gives:
\[ x^2 + x + 1 = 0 \]

### Step 8: Use the quadratic formula
Now, apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1, b = 1, c = 1\):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 - 4}}{2} \]
\[ x = \frac{-1 \pm \sqrt{-3}}{2} \]
\[ x = \frac{-1 \pm i\sqrt{3}}{2} \]

### Step 9: Find \(x^3\)
Now, we need to find \(x^3\):
\[ x^3 = \left(\frac{-1 \pm i\sqrt{3}}{2}\right)^3 \]
Using the binomial expansion or direct calculation, we find:
\[ x^3 = -1 \]

### Conclusion
Thus, the value of \(\left(\frac{r}{s}\right)^3\) is:
\[ \left(\frac{r}{s}\right)^3 = -1 \]