1. **State the problem:** Simplify the expression $$\frac{(q r p^{-2})^{-2} p^{-2} q^{5} r^{0}}{q r}$$ and verify if the result is $$\frac{1}{q^{4} r^{3} p^{6}}$$. 2. **Recall the rules:** - Power of a power: $$(a^{m})^{n} = a^{m \times n}$$ - Negative exponents: $$a^{-m} = \frac{1}{a^{m}}$$ - Multiplying powers with the same base: $$a^{m} \times a^{n} = a^{m+n}$$ - Dividing powers with the same base: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$ - Any number to the zero power is 1: $$a^{0} = 1$$ 3. **Simplify the numerator:** $$(q r p^{-2})^{-2} p^{-2} q^{5} r^{0} = q^{-2 \times 1} r^{-2 \times 1} p^{-2 \times (-2)} p^{-2} q^{5} \times 1$$ Calculate powers inside: $$= q^{-2} r^{-2} p^{4} p^{-2} q^{5}$$ 4. **Combine like bases in numerator:** $$= q^{-2 + 5} r^{-2} p^{4 - 2} = q^{3} r^{-2} p^{2}$$ 5. **Write the entire expression:** $$\frac{q^{3} r^{-2} p^{2}}{q^{1} r^{1}}$$ 6. **Divide powers with the same base:** $$= q^{3 - 1} r^{-2 - 1} p^{2} = q^{2} r^{-3} p^{2}$$ 7. **Rewrite negative exponents as fractions:** $$= q^{2} \times \frac{1}{r^{3}} \times p^{2} = \frac{q^{2} p^{2}}{r^{3}}$$ 8. **Final simplified form:** $$\frac{q^{2} p^{2}}{r^{3}}$$ **Compare with your answer:** Your answer was $$\frac{1}{q^{4} r^{3} p^{6}}$$ which is different from the correct simplification. **Therefore, the correct simplified expression is:** $$\boxed{\frac{q^{2} p^{2}}{r^{3}}}$$