Question: Simplify the expression $$\frac{2x^4y}{3x^2} \cdot \frac{4xy^3}{2x^3y^2}$$
1. **State the problem:** Simplify the expression $$\frac{2x^4y}{3x^2} \cdot \frac{4xy^3}{2x^3y^2}$$.
2. **Write the formula and rules:** When multiplying fractions, multiply the numerators together and the denominators together:
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$
Also, use the laws of exponents: $$x^m \cdot x^n = x^{m+n}$$ and $$\frac{x^m}{x^n} = x^{m-n}$$.
3. **Multiply the numerators and denominators:**
$$\frac{2x^4y}{3x^2} \cdot \frac{4xy^3}{2x^3y^2} = \frac{2x^4y \cdot 4xy^3}{3x^2 \cdot 2x^3y^2} = \frac{8x^{4+1}y^{1+3}}{6x^{2+3}y^2} = \frac{8x^5y^4}{6x^5y^2}$$
4. **Simplify the fraction:**
Cancel common factors in numerator and denominator:
$$\frac{\cancel{8}x^5y^4}{\cancel{6}x^5y^2} = \frac{4x^5y^4}{3x^5y^2}$$
Now cancel $$x^5$$:
$$\frac{4\cancel{x^5}y^4}{3\cancel{x^5}y^2} = \frac{4y^4}{3y^2}$$
5. **Simplify the powers of $$y$$:**
$$\frac{4y^4}{3y^2} = \frac{4}{3} y^{4-2} = \frac{4}{3} y^2$$
6. **Final answer:**
$$\boxed{\frac{4}{3} y^2}$$