1. **State the problem:** Simplify the expression $$\frac{10(p^3 q^2 r^0)^{-3}}{(8p^{-3} q^5 r^3)^{-2}}$$. 2. **Recall the rules:** - Any term raised to zero is 1, so $r^0 = 1$. - Power of a power: $(a^m)^n = a^{mn}$. - Negative exponents: $a^{-m} = \frac{1}{a^m}$. - When dividing powers with the same base, subtract exponents. 3. **Simplify inside the parentheses:** - Since $r^0 = 1$, rewrite numerator term as $(p^3 q^2)^ {-3}$. 4. **Apply the power to each factor:** $$ (p^3 q^2)^{-3} = p^{3 \times (-3)} q^{2 \times (-3)} = p^{-9} q^{-6} $$ 5. **Simplify denominator term:** $$ (8 p^{-3} q^5 r^3)^{-2} = 8^{-2} p^{-3 \times (-2)} q^{5 \times (-2)} r^{3 \times (-2)} = 8^{-2} p^{6} q^{-10} r^{-6} $$ 6. **Rewrite the entire expression:** $$ \frac{10 p^{-9} q^{-6}}{8^{-2} p^{6} q^{-10} r^{-6}} $$ 7. **Simplify constants:** $$ 8^{-2} = \frac{1}{8^2} = \frac{1}{64} $$ So denominator constant is $\frac{1}{64}$, which moves to numerator as $64$: $$ 10 p^{-9} q^{-6} \times 64 p^{-6} q^{10} r^{6} $$ 8. **Combine constants:** $$ 10 \times 64 = 640 $$ 9. **Combine powers of $p$:** $$ p^{-9} \times p^{6} = p^{-9+6} = p^{-3} $$ 10. **Combine powers of $q$:** $$ q^{-6} \times q^{10} = q^{-6+10} = q^{4} $$ 11. **Include $r^{6}$:** 12. **Final simplified expression:** $$ 640 p^{-3} q^{4} r^{6} = \frac{640 q^{4} r^{6}}{p^{3}} $$ **Answer:** $$\boxed{\frac{640 q^{4} r^{6}}{p^{3}}}$$