1. **State the problem:** Simplify the expression $$\frac{3a^9 r^{-3}}{-4 (a^2)^{-1}}$$. 2. **Recall the rules:** - When you have a power raised to another power, multiply the exponents: $$(a^m)^n = a^{mn}$$. - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^n}$$. - When dividing powers with the same base, subtract exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Simplify the denominator:** $$(a^2)^{-1} = a^{2 \times (-1)} = a^{-2} = \frac{1}{a^2}$$. 4. **Rewrite the expression:** $$\frac{3a^9 r^{-3}}{-4 a^{-2}}$$. 5. **Combine the powers of $a$ in numerator and denominator:** $$\frac{3a^9 r^{-3}}{-4 a^{-2}} = \frac{3 r^{-3} a^9}{-4 a^{-2}} = \frac{3 r^{-3} \cancel{a^9}}{-4 \cancel{a^{-2}}} = \frac{3 r^{-3} a^{9 - (-2)}}{-4} = \frac{3 r^{-3} a^{11}}{-4}$$. 6. **Rewrite $r^{-3}$ as a fraction:** $$r^{-3} = \frac{1}{r^3}$$. 7. **Final simplified expression:** $$\frac{3 a^{11}}{-4 r^3} = -\frac{3 a^{11}}{4 r^3}$$. **Answer:** $$-\frac{3 a^{11}}{4 r^3}$$