1. **Simplify** $\frac{6y}{x} \div \frac{2}{3xy^2}$. 2. Division of fractions means multiplying by the reciprocal: $$\frac{6y}{x} \times \frac{3xy^2}{2}$$ 3. Multiply numerators and denominators: $$\frac{6y \times 3xy^2}{x \times 2} = \frac{18xy^3}{2x}$$ 4. Cancel common factor $x$ in numerator and denominator: $$\frac{18\cancel{x}y^3}{2\cancel{x}} = \frac{18y^3}{2}$$ 5. Simplify the fraction by dividing numerator and denominator by 2: $$\frac{\cancel{18}^{9}y^3}{\cancel{2}^1} = 9y^3$$ **Final answer:** $9y^3$ --- **Problem:** Simplify $\frac{6y}{x} \div \frac{2}{3xy^2}$. **Formula:** Division of fractions $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$. **Explanation:** When dividing fractions, multiply by the reciprocal of the divisor. **Intermediate work:** - Multiply: $\frac{6y}{x} \times \frac{3xy^2}{2} = \frac{18xy^3}{2x}$ - Cancel $x$: $\frac{18\cancel{x}y^3}{2\cancel{x}} = \frac{18y^3}{2}$ - Simplify fraction: $\frac{\cancel{18}^{9}y^3}{\cancel{2}^1} = 9y^3$ --- "slug": "fraction division", "subject": "algebra", "desmos": {"latex": "y=9x^3", "features": {"intercepts": true, "extrema": true}}, "q_count": 3