1. **State the problem:** Simplify the expression $$\frac{(3a^{2}b^{3}c^{-2})^{-2}}{a^{2}b^{-1}c^{4}}$$. 2. **Recall the rules:** - When raising a power to another power, multiply the exponents: $$(x^{m})^{n} = x^{mn}$$. - Negative exponents mean reciprocal: $$x^{-m} = \frac{1}{x^{m}}$$. - When dividing like bases, subtract exponents: $$\frac{x^{m}}{x^{n}} = x^{m-n}$$. 3. **Apply the power to the numerator:** $$ (3a^{2}b^{3}c^{-2})^{-2} = 3^{-2}a^{2 \times (-2)}b^{3 \times (-2)}c^{-2 \times (-2)} = 3^{-2}a^{-4}b^{-6}c^{4} $$ 4. **Rewrite the entire expression:** $$ \frac{3^{-2}a^{-4}b^{-6}c^{4}}{a^{2}b^{-1}c^{4}} $$ 5. **Divide the terms with the same base by subtracting exponents:** $$ 3^{-2}a^{-4 - 2}b^{-6 - (-1)}c^{4 - 4} = 3^{-2}a^{-6}b^{-5}c^{0} $$ 6. **Simplify powers:** Since $$c^{0} = 1$$, the expression becomes: $$ 3^{-2}a^{-6}b^{-5} $$ 7. **Rewrite negative exponents as positive by taking reciprocals:** $$ \frac{1}{3^{2}a^{6}b^{5}} = \frac{1}{9a^{6}b^{5}} $$ **Final answer:** $$ \boxed{\frac{1}{9a^{6}b^{5}}} $$