1. **Problem 10:** Simplify $ \frac{a^3b^3}{xy^4} \div \frac{a^2b}{x^2y} $. 2. Division of fractions means multiplying by the reciprocal: $$ \frac{a^3b^3}{xy^4} \times \frac{x^2y}{a^2b} $$ 3. Multiply numerators and denominators: $$ \frac{a^3b^3 \cdot x^2 y}{x y^4 \cdot a^2 b} $$ 4. Group like terms: $$ \frac{a^{3} b^{3} x^{2} y^{1}}{a^{2} b^{1} x^{1} y^{4}} $$ 5. Simplify by subtracting exponents: $$ a^{3-2} b^{3-1} x^{2-1} y^{1-4} = a^{1} b^{2} x^{1} y^{-3} $$ 6. Rewrite negative exponent: $$ a b^{2} x \frac{1}{y^{3}} = \frac{a b^{2} x}{y^{3}} $$ --- 7. **Problem 11:** Simplify $$ \frac{\frac{4x}{x+6}}{\frac{x^{2} - 3x}{x^{2} + 3x - 18}} $$ 8. Division of fractions means multiply by reciprocal: $$ \frac{4x}{x+6} \times \frac{x^{2} + 3x - 18}{x^{2} - 3x} $$ 9. Factor polynomials: - $ x^{2} + 3x - 18 = (x+6)(x-3) $ - $ x^{2} - 3x = x(x-3) $ 10. Substitute factors: $$ \frac{4x}{x+6} \times \frac{(x+6)(x-3)}{x(x-3)} $$ 11. Cancel common factors $ (x+6) $ and $ (x-3) $: $$ \frac{4x}{\cancel{x+6}} \times \frac{\cancel{x+6} \cancel{(x-3)}}{x \cancel{(x-3)}} $$ 12. Simplify: $$ \frac{4x}{1} \times \frac{1}{x} $$ 13. Cancel $ x $: $$ \frac{\cancel{4x}}{1} \times \frac{1}{\cancel{x}} = 4 $$ **Final answers:** - Problem 10: $$ \frac{a b^{2} x}{y^{3}} $$ - Problem 11: $$ 4 $$