Simplify the given expressions
`(a^(p+q))^(p-q) (a^(q+r))^(q-r) (a^(r+p))^(r-p)`

Answer

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

### Step-by-Step Solution:

1. **Apply the Power of a Power Property**:
We know that \((x^m)^n = x^{m \cdot n}\). Therefore, we can rewrite each term:
\[ (a^{(p+q)})^{(p-q)} = a^{(p+q)(p-q)} \]
\[ (a^{(q+r)})^{(q-r)} = a^{(q+r)(q-r)} \]
\[ (a^{(r+p)})^{(r-p)} = a^{(r+p)(r-p)} \]

2. **Combine the Exponents**:
Now we can combine these into a single expression:
\[ a^{(p+q)(p-q)} \cdot a^{(q+r)(q-r)} \cdot a^{(r+p)(r-p)} = a^{(p+q)(p-q) + (q+r)(q-r) + (r+p)(r-p)} \]

3. **Expand Each Product**:
Now we will expand each of the products in the exponent:
- For \((p+q)(p-q)\):
\[ (p+q)(p-q) = p^2 - q^2 \]
- For \((q+r)(q-r)\):
\[ (q+r)(q-r) = q^2 - r^2 \]
- For \((r+p)(r-p)\):
\[ (r+p)(r-p) = r^2 - p^2 \]

4. **Combine the Expanded Terms**:
Now we can substitute these back into the exponent:
\[ a^{(p^2 - q^2) + (q^2 - r^2) + (r^2 - p^2)} \]

5. **Simplify the Exponent**:
Combine the terms in the exponent:
\[ p^2 - q^2 + q^2 - r^2 + r^2 - p^2 = 0 \]

6. **Final Result**:
Since we have \(a^0\), we know that any non-zero number raised to the power of zero is 1:
\[ a^0 = 1 \]

### Conclusion:
The simplified expression is:
\[ \boxed{1} \]