1. **State the problem:** Simplify the expression $$\frac{x^3}{2y^2} \cdot \frac{6y^4}{xy}$$. 2. **Write the expression:** $$\frac{x^3}{2y^2} \times \frac{6y^4}{xy}$$. 3. **Multiply the numerators and denominators:** $$\frac{x^3 \times 6y^4}{2y^2 \times xy} = \frac{6x^3 y^4}{2x y^3}$$. 4. **Simplify coefficients:** $$\frac{6}{2} = 3$$, so expression becomes $$\frac{3x^3 y^4}{x y^3}$$. 5. **Simplify variables:** - For $x$: $$\frac{x^3}{x} = x^{3-1} = x^2$$. - For $y$: $$\frac{y^4}{y^3} = y^{4-3} = y$$. 6. **Final simplified expression:** $$3x^2 y$$. Therefore, the simplified form of the given expression is $$3x^2 y$$.