The value of \( x^{p-q} x^{q-r} x^{r-p} \) is equal to \( \begin{array}{ll}\text { a) } x^{p q r} & \text { b) } 0 \\ \text { c) } x & \text { d) } 1\end{array} \)

To evaluate the expression \( x^{p-q} \cdot x^{q-r} \cdot x^{r-p} \), we can use the properties of exponents: 1. **Multiplication of Like Bases:** When multiplying expressions with the same base, you add the exponents: \[ x^{a} \cdot x^{b} \cdot x^{c} = x^{a+b+c} \] 2. **Applying to the Given Expression:** \[ x^{p-q} \cdot x^{q-r} \cdot x^{r-p} = x^{(p - q) + (q - r) + (r - p)} \] 3. **Simplifying the Exponents:** \[ (p - q) + (q - r) + (r - p) = p - q + q - r + r - p \] Notice that each term cancels out: \[ p - p + q - q + r - r = 0 \] 4. **Final Expression:** \[ x^{0} = 1 \quad \text{(assuming } x \neq 0\text{)} \] **Answer:** \( \boxed{1} \)