Given `pq + qr + rp = 0` find
`(1)/(p^(2)-qr) + (1)/(q^(2) -rp) + (1)/(r^(2) - pq)`= ?

Answer

To solve the equation given \( pq + qr + rp = 0 \) and find the value of

\[ \frac{1}{p^2 - qr} + \frac{1}{q^2 - rp} + \frac{1}{r^2 - pq}, \]

we will follow these steps:

### Step 1: Rewrite the terms using \( pq + qr + rp = 0 \)

From the equation \( pq + qr + rp = 0 \), we can express \( qr \), \( rp \), and \( pq \) in terms of the other variables:

- \( qr = - (pq + rp) \)
- \( rp = - (pq + qr) \)
- \( pq = - (qr + rp) \)

### Step 2: Substitute the values into the expression

Now we can substitute these values into the expression:

1. For \( \frac{1}{p^2 - qr} \):
\[ \frac{1}{p^2 - qr} = \frac{1}{p^2 + (pq + rp)} = \frac{1}{p^2 + pq + rp} \]

2. For \( \frac{1}{q^2 - rp} \):
\[ \frac{1}{q^2 - rp} = \frac{1}{q^2 + (pq + qr)} = \frac{1}{q^2 + pq + qr} \]

3. For \( \frac{1}{r^2 - pq} \):
\[ \frac{1}{r^2 - pq} = \frac{1}{r^2 + (qr + rp)} = \frac{1}{r^2 + qr + rp} \]

### Step 3: Combine the terms

Now we can combine these fractions:

\[ \frac{1}{p^2 + pq + rp} + \frac{1}{q^2 + pq + qr} + \frac{1}{r^2 + qr + rp} \]

### Step 4: Simplify the expression

Notice that \( pq + qr + rp = 0 \) implies:

- \( p^2 + pq + rp = p^2 - (qr) \)
- \( q^2 + pq + qr = q^2 - (rp) \)
- \( r^2 + qr + rp = r^2 - (pq) \)

Thus, we can rewrite the expression as:

\[ \frac{1}{p^2 - qr} + \frac{1}{q^2 - rp} + \frac{1}{r^2 - pq} \]

### Step 5: Final Result

Since \( pq + qr + rp = 0 \), we can conclude that each of the terms simplifies to a common form, leading to the final result:

\[ \frac{1}{p^2 - qr} + \frac{1}{q^2 - rp} + \frac{1}{r^2 - pq} = 0 \]

### Final Answer:

\[ \boxed{0} \]