1. **State the problem:** Simplify the expression $$\frac{(4x^{2}y)(2x^{3}y)}{(2x)^{3}(2y)^{2}}$$. 2. **Write the expression clearly:** $$\frac{4x^{2}y \times 2x^{3}y}{(2x)^{3} \times (2y)^{2}}$$ 3. **Multiply the numerator terms:** $$4 \times 2 = 8$$ $$x^{2} \times x^{3} = x^{2+3} = x^{5}$$ $$y \times y = y^{2}$$ So numerator becomes: $$8x^{5}y^{2}$$ 4. **Expand the denominator powers:** $$(2x)^{3} = 2^{3} x^{3} = 8x^{3}$$ $$(2y)^{2} = 2^{2} y^{2} = 4y^{2}$$ So denominator becomes: $$8x^{3} \times 4y^{2} = 32x^{3}y^{2}$$ 5. **Rewrite the fraction:** $$\frac{8x^{5}y^{2}}{32x^{3}y^{2}}$$ 6. **Simplify the fraction coefficients:** $$\frac{\cancel{8}x^{5}y^{2}}{\cancel{32}x^{3}y^{2}} = \frac{1x^{5}y^{2}}{4x^{3}y^{2}}$$ 7. **Simplify the variables:** $$x^{5} \div x^{3} = x^{5-3} = x^{2}$$ $$y^{2} \div y^{2} = y^{2-2} = y^{0} = 1$$ 8. **Final simplified expression:** $$\frac{x^{2}}{4}$$ **Answer:** $$\frac{x^{2}}{4}$$