If pqr = 1, then what is value of the expression `(1)/(1+p+q^(-1)) +(1)/(1+q+r^(-1)) +(1)/(1+r+p^(-1))` ?

Options

A. 1
B. `-1`
C. 0
D. `(1)/(3)`

Answer

To solve the expression

\[ \frac{1}{1 + p + q^{-1}} + \frac{1}{1 + q + r^{-1}} + \frac{1}{1 + r + p^{-1}} \]

given that \( pqr = 1 \), we can follow these steps:

### Step 1: Rewrite the expression
First, we rewrite each term in the expression using the property that \( r = \frac{1}{pq} \) since \( pqr = 1 \).

1. The first term becomes:
\[ \frac{1}{1 + p + q^{-1}} = \frac{1}{1 + p + \frac{1}{q}} = \frac{1}{1 + p + \frac{1}{q}} = \frac{q}{q + pq + 1} \]

### Step 2: Simplify the second term
For the second term:
\[ \frac{1}{1 + q + r^{-1}} = \frac{1}{1 + q + \frac{1}{r}} = \frac{1}{1 + q + pq} = \frac{1}{1 + q + pq} \]

### Step 3: Simplify the third term
For the third term:
\[ \frac{1}{1 + r + p^{-1}} = \frac{1}{1 + \frac{1}{pq} + \frac{1}{p}} = \frac{1}{1 + \frac{1}{pq} + \frac{1}{p}} = \frac{pq}{pq + 1 + q} \]

### Step 4: Combine the terms
Now we need to combine all three terms:
\[ x + y + z = \frac{q}{q + pq + 1} + \frac{1}{1 + q + pq} + \frac{pq}{pq + 1 + q} \]

### Step 5: Find a common denominator
The common denominator for all three fractions is \( (1 + q + pq) \). Therefore, we can write:
\[ x + y + z = \frac{q + 1 + pq}{1 + q + pq} \]

### Step 6: Simplify the expression
Notice that the numerator simplifies to:
\[ q + 1 + pq = 1 + q + pq \]
Thus, we have:
\[ x + y + z = \frac{1 + q + pq}{1 + q + pq} = 1 \]

### Conclusion
The value of the expression is:
\[ \boxed{1} \]