1. **State the problem:** Simplify the expression $$(4a^3 b^6 c^{-2})^{-3}$$. 2. **Recall the power of a product rule:** $$(xyz)^n = x^n y^n z^n$$ and the power of a power rule: $$(x^m)^n = x^{mn}$$. 3. **Apply the power of a product rule:** $$ (4a^3 b^6 c^{-2})^{-3} = 4^{-3} (a^3)^{-3} (b^6)^{-3} (c^{-2})^{-3} $$ 4. **Apply the power of a power rule to each term:** $$ 4^{-3} = \frac{1}{4^3} $$ $$ (a^3)^{-3} = a^{3 \times (-3)} = a^{-9} $$ $$ (b^6)^{-3} = b^{6 \times (-3)} = b^{-18} $$ $$ (c^{-2})^{-3} = c^{-2 \times (-3)} = c^{6} $$ 5. **Combine all terms:** $$ \frac{1}{4^3} \times a^{-9} \times b^{-18} \times c^{6} = \frac{c^{6}}{4^3 a^{9} b^{18}} $$ 6. **Rewrite with positive exponents only:** $$ \frac{c^{6}}{4^3 a^{9} b^{18}} $$ 7. **Compare with options:** - Option D is $$\frac{a^{9} b^{12} c^{6}}{4^{3}}$$ which is close but powers of $b$ differ. - None of the options exactly match the simplified expression. **Final simplified expression:** $$ \boxed{\frac{c^{6}}{4^{3} a^{9} b^{18}}} $$