1. **State the problem:** Simplify the expression $$\left(\frac{-4 a^4 b^5 c^0}{2 a^2 b^3 c^6}\right)^{-3}$$. 2. **Recall the rules:** - Any number or variable raised to the zero power is 1, so $c^0 = 1$. - When dividing powers with the same base, subtract the exponents: $\frac{x^m}{x^n} = x^{m-n}$. - A negative exponent means take the reciprocal: $x^{-n} = \frac{1}{x^n}$. 3. **Simplify inside the parentheses first:** $$\frac{-4 a^4 b^5 c^0}{2 a^2 b^3 c^6} = \frac{-4 a^4 b^5 \times 1}{2 a^2 b^3 c^6} = \frac{-4 a^4 b^5}{2 a^2 b^3 c^6}$$ 4. **Divide coefficients and variables separately:** $$\frac{-4}{2} = -2$$ $$\frac{a^4}{a^2} = a^{4-2} = a^2$$ $$\frac{b^5}{b^3} = b^{5-3} = b^2$$ $$\frac{1}{c^6} = c^{-6}$$ So the expression inside the parentheses is: $$-2 a^2 b^2 c^{-6}$$ 5. **Apply the negative exponent -3 to the entire expression:** $$\left(-2 a^2 b^2 c^{-6}\right)^{-3} = \frac{1}{\left(-2 a^2 b^2 c^{-6}\right)^3}$$ 6. **Raise each factor to the power 3:** $$(-2)^3 = -8$$ $$\left(a^2\right)^3 = a^{2 \times 3} = a^6$$ $$\left(b^2\right)^3 = b^{2 \times 3} = b^6$$ $$\left(c^{-6}\right)^3 = c^{-6 \times 3} = c^{-18}$$ 7. **Put it all together:** $$\frac{1}{-8 a^6 b^6 c^{-18}} = \frac{1}{-8 a^6 b^6} \times c^{18} = \frac{c^{18}}{-8 a^6 b^6}$$ **Final answer:** $$\boxed{\frac{c^{18}}{-8 a^6 b^6}}$$