1. **State the problem:** Simplify the expression $$\frac{6y}{x} \div \frac{2}{3xy^2}$$. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{6y}{x} \div \frac{2}{3xy^2} = \frac{6y}{x} \times \frac{3xy^2}{2}$$. 3. **Multiply the numerators and denominators:** $$= \frac{6y \times 3xy^2}{x \times 2} = \frac{18xy^3}{2x}$$. 4. **Simplify the fraction:** Cancel common factors in numerator and denominator. $$= \frac{\cancel{18} \times \cancel{x} y^3}{2 \times \cancel{x}} = \frac{18 y^3}{2}$$. 5. **Simplify the numeric fraction:** $$= 9 y^3$$. **Final answer:** $$9 y^3$$.