1. Stating the problem: Simplify the expression $$\frac{10 x^6}{4 y^2} \times \frac{-4 a^4 b^5 c^0}{2 a^2 b^3 c^6} \times (-3) \times 9^{-2}$$. 2. Recall the rules: - When multiplying fractions, multiply numerators and denominators separately. - For powers with the same base, use $a^m \times a^n = a^{m+n}$. - For division of powers with the same base, use $\frac{a^m}{a^n} = a^{m-n}$. - Any number to the zero power is 1, so $c^0 = 1$. - Negative exponents mean reciprocal: $a^{-n} = \frac{1}{a^n}$. 3. Simplify each part step-by-step: First, multiply the fractions: $$\frac{10 x^6}{4 y^2} \times \frac{-4 a^4 b^5 c^0}{2 a^2 b^3 c^6} = \frac{10 x^6 \times (-4) a^4 b^5 \times 1}{4 y^2 \times 2 a^2 b^3 c^6} = \frac{-40 x^6 a^4 b^5}{8 y^2 a^2 b^3 c^6}$$ 4. Simplify the coefficients: $$\frac{-40}{8} = \cancel{\frac{-40}{8}} = -5$$ 5. Simplify the variables using exponent rules: - For $a$: $a^{4} / a^{2} = a^{4-2} = a^{2}$ - For $b$: $b^{5} / b^{3} = b^{5-3} = b^{2}$ - For $c$: $c^{0} / c^{6} = c^{0-6} = c^{-6}$ So the fraction becomes: $$-5 x^{6} a^{2} b^{2} \frac{1}{y^{2} c^{6}}$$ 6. Now multiply by $-3$: $$-5 x^{6} a^{2} b^{2} \frac{1}{y^{2} c^{6}} \times (-3) = 15 x^{6} a^{2} b^{2} \frac{1}{y^{2} c^{6}}$$ 7. Finally, multiply by $9^{-2}$: Recall $9^{-2} = \frac{1}{9^{2}} = \frac{1}{81}$. So: $$15 x^{6} a^{2} b^{2} \frac{1}{y^{2} c^{6}} \times \frac{1}{81} = \frac{15}{81} x^{6} a^{2} b^{2} \frac{1}{y^{2} c^{6}}$$ 8. Simplify the coefficient fraction: $$\frac{15}{81} = \frac{5}{27}$$ 9. Final simplified expression: $$\frac{5 x^{6} a^{2} b^{2}}{27 y^{2} c^{6}}$$