1. **State the problem:** Simplify the expression $$\frac{(2 a^3 b^2 c^{-2})^3}{16} a^{-4} b^5 c^{-6}$$. 2. **Apply the power to each factor inside the parentheses:** $$(2 a^3 b^2 c^{-2})^3 = 2^3 (a^3)^3 (b^2)^3 (c^{-2})^3 = 8 a^{9} b^{6} c^{-6}$$ 3. **Rewrite the expression substituting this result:** $$\frac{8 a^{9} b^{6} c^{-6}}{16} a^{-4} b^{5} c^{-6}$$ 4. **Simplify the fraction:** $$\frac{8}{16} = \frac{\cancel{8}}{2 \times \cancel{8}} = \frac{1}{2}$$ So the expression becomes: $$\frac{1}{2} a^{9} b^{6} c^{-6} a^{-4} b^{5} c^{-6}$$ 5. **Combine like bases by adding exponents:** - For $a$: $a^{9} \times a^{-4} = a^{9 + (-4)} = a^{5}$ - For $b$: $b^{6} \times b^{5} = b^{6 + 5} = b^{11}$ - For $c$: $c^{-6} \times c^{-6} = c^{-6 + (-6)} = c^{-12}$ 6. **Write the simplified expression:** $$\frac{1}{2} a^{5} b^{11} c^{-12}$$ 7. **Optional: rewrite negative exponent as positive exponent in denominator:** $$\frac{a^{5} b^{11}}{2 c^{12}}$$ **Final answer:** $$\frac{a^{5} b^{11}}{2 c^{12}}$$