If ` (1)/( p) + (1)/(q) = (1)/( p + q)` then the value of ` (p^(3) + q^(3)) ` is

Options

A. p - q
B. pq
C. 1
D. 0

Answer

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

### Step 1: Rewrite the equation
Start with the given equation:
\[ \frac{1}{p} + \frac{1}{q} = \frac{1}{p + q} \]

### Step 2: Find a common denominator
The left-hand side can be rewritten with a common denominator:
\[ \frac{q + p}{pq} = \frac{1}{p + q} \]

### Step 3: Cross-multiply
Cross-multiplying gives us:
\[ (q + p)(p + q) = pq \]

### Step 4: Expand the left-hand side
Expanding the left-hand side results in:
\[ p^2 + q^2 + 2pq = pq \]

### Step 5: Rearrange the equation
Rearranging the equation leads to:
\[ p^2 + q^2 + 2pq - pq = 0 \]
This simplifies to:
\[ p^2 + q^2 + pq = 0 \]

### Step 6: Use the identity for \( p^3 + q^3 \)
We know the identity:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]
From Step 5, we have \( p^2 + q^2 = -pq \). Therefore:
\[ p^2 - pq + q^2 = -pq - pq = -2pq \]

### Step 7: Substitute into the identity
Now substituting back into the identity gives:
\[ p^3 + q^3 = (p + q)(-2pq) \]

### Step 8: Evaluate \( p + q \) and \( pq \)
From the previous steps, we see that \( pq = 0 \) (since \( p^2 + q^2 + pq = 0 \)). Thus:
\[ p^3 + q^3 = (p + q)(-2 \cdot 0) = 0 \]

### Final Answer
The value of \( p^3 + q^3 \) is:
\[ \boxed{0} \]