Find the value of `(a^(p)/a^(q))^(p+q-r)xx(a^(r)/a^(p))^(r+p-q)xx(a^(q)/a^(r))^(q+r-p)`

Options

A. `a^(pqr)`
B. `a^(p+q+r)`
C. `a^(pq+qr+pr)`
D. 1

Answer

To solve the expression \((\frac{a^p}{a^q})^{p+q-r} \times (\frac{a^r}{a^p})^{r+p-q} \times (\frac{a^q}{a^r})^{q+r-p}\), we will follow these steps:

### Step 1: Simplify Each Fraction
Using the property of indices \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify each fraction:
1. \(\frac{a^p}{a^q} = a^{p-q}\)
2. \(\frac{a^r}{a^p} = a^{r-p}\)
3. \(\frac{a^q}{a^r} = a^{q-r}\)

### Step 2: Rewrite the Expression
Now, we can rewrite the original expression using the simplified fractions:
\[ (a^{p-q})^{p+q-r} \times (a^{r-p})^{r+p-q} \times (a^{q-r})^{q+r-p} \]

### Step 3: Apply the Power of a Power Rule
Using the property \((a^m)^n = a^{m \cdot n}\), we can further simplify:
1. \((a^{p-q})^{p+q-r} = a^{(p-q)(p+q-r)}\)
2. \((a^{r-p})^{r+p-q} = a^{(r-p)(r+p-q)}\)
3. \((a^{q-r})^{q+r-p} = a^{(q-r)(q+r-p)}\)

### Step 4: Combine the Exponents
Now, we can combine the expressions since they have the same base \(a\):
\[ a^{(p-q)(p+q-r) + (r-p)(r+p-q) + (q-r)(q+r-p)} \]

### Step 5: Expand Each Term
Now we will expand each term:
1. \((p-q)(p+q-r) = p^2 + pq - pr - qp - q^2 + qr\)
2. \((r-p)(r+p-q) = r^2 + rp - rq - pr - p^2 + pq\)
3. \((q-r)(q+r-p) = q^2 + qr - qp - rq - r^2 + rp\)

### Step 6: Combine All the Expanded Terms
Now we combine all the expanded terms:
\[ p^2 + pq - pr - qp - q^2 + qr + r^2 + rp - rq - pr - p^2 + pq + q^2 + qr - qp - rq - r^2 + rp \]

### Step 7: Simplify the Combined Expression
Now we simplify the combined expression:
- \(p^2\) and \(-p^2\) cancel out.
- \(q^2\) and \(-q^2\) cancel out.
- \(r^2\) and \(-r^2\) cancel out.
- The remaining terms will cancel out as well.

### Final Result
After all cancellations, we find that:
\[ a^0 = 1 \]

Thus, the value of the original expression is **1**.