On simplification `(a^p/a^q)^(p+q) xx(a^q/a^r)^(q+r)xx(a^r/a^p)^(r+p)` yeilds

Options

A. 1. `a^(p+q+r)`
B. 2. `a^(pq+qr+rp)`
C. 3. 0
D. 4. 1

Answer

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

### Step 1: Apply the Quotient Rule of Indices
Using the property of indices that states \(\frac{a^m}{a^n} = a^{m-n}\), we can rewrite each fraction in the expression:

\[ \frac{a^p}{a^q} = a^{p-q}, \quad \frac{a^q}{a^r} = a^{q-r}, \quad \frac{a^r}{a^p} = a^{r-p} \]

### Step 2: Substitute Back into the Expression
Now we substitute these back into the original expression:

\[ (a^{p-q})^{(p+q)} \times (a^{q-r})^{(q+r)} \times (a^{r-p})^{(r+p)} \]

### Step 3: Apply the Power Rule of Indices
Using the property of indices that states \((a^m)^n = a^{m \cdot n}\), we can simplify each term:

\[ a^{(p-q)(p+q)} \times a^{(q-r)(q+r)} \times a^{(r-p)(r+p)} \]

### Step 4: Combine the Exponents
Since the bases are the same, we can add the exponents:

\[ a^{(p-q)(p+q) + (q-r)(q+r) + (r-p)(r+p)} \]

### Step 5: Expand Each Term
Now we expand each of the products in the exponent:

1. \((p-q)(p+q) = p^2 - q^2\)
2. \((q-r)(q+r) = q^2 - r^2\)
3. \((r-p)(r+p) = r^2 - p^2\)

Putting these together, we have:

\[ p^2 - q^2 + q^2 - r^2 + r^2 - p^2 \]

### Step 6: Simplify the Exponents
Notice that \(p^2\) and \(-p^2\) cancel out, \(q^2\) and \(-q^2\) cancel out, and \(r^2\) and \(-r^2\) cancel out:

\[ 0 \]

### Step 7: Final Result
Thus, we have:

\[ a^0 = 1 \]

So the final answer is:

\[ \boxed{1} \]

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