If `pq (p +q) =1,` then the value of`(1)/( p ^(3) q ^(3)) - p ^(3) - q ^(3)` is equal to

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we start with the given equation:

\[ pq(p + q) = 1 \]

We need to find the value of:

\[ \frac{1}{p^3 q^3} - p^3 - q^3 \]

### Step 1: Rewrite the expression
First, we can rewrite the expression we need to evaluate:

\[ \frac{1}{p^3 q^3} - p^3 - q^3 = \frac{1}{(pq)^3} - p^3 - q^3 \]

### Step 2: Use the given equation
From the given equation \( pq(p + q) = 1 \), we can express \( p + q \):

\[ p + q = \frac{1}{pq} \]

### Step 3: Substitute \( p + q \) into the expression
Now, we can substitute \( p + q \) into our expression. We know that:

\[ (p + q)^3 = p^3 + q^3 + 3pq(p + q) \]

Using \( p + q = \frac{1}{pq} \):

\[ \left(\frac{1}{pq}\right)^3 = p^3 + q^3 + 3pq\left(\frac{1}{pq}\right) \]

### Step 4: Simplify the equation
This simplifies to:

\[ \frac{1}{(pq)^3} = p^3 + q^3 + 3 \]

### Step 5: Rearranging the equation
Now, we can rearrange this to find \( p^3 + q^3 \):

\[ p^3 + q^3 = \frac{1}{(pq)^3} - 3 \]

### Step 6: Substitute back into the original expression
Now, we substitute \( p^3 + q^3 \) back into our original expression:

\[ \frac{1}{p^3 q^3} - p^3 - q^3 = \frac{1}{(pq)^3} - \left(\frac{1}{(pq)^3} - 3\right) \]

### Step 7: Simplify the expression
This simplifies to:

\[ \frac{1}{(pq)^3} - \frac{1}{(pq)^3} + 3 = 3 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{3} \]