1. **State the problem:** Simplify the expression $\left(\frac{75p^{3}m^{-5}}{15p^{5}m^{-4}r^{-8}}\right)^0$. 2. **Recall the rule:** Any nonzero expression raised to the power of 0 equals 1, i.e., $a^0 = 1$ for $a \neq 0$. 3. **Apply the rule:** Since the entire fraction is raised to the 0 power, the value is 1 regardless of the fraction's contents. 4. **Final answer:** $$\left(\frac{75p^{3}m^{-5}}{15p^{5}m^{-4}r^{-8}}\right)^0 = 1$$ Note: The second expression is incomplete and cannot be fully simplified.