1. **State the problem:** Simplify the expression $$\frac{(2x)^2}{(3x^2Y)^3}$$. 2. **Apply the power to each factor inside the parentheses:** $$(2x)^2 = 2^2 \cdot x^2 = 4x^2$$ $$(3x^2Y)^3 = 3^3 \cdot (x^2)^3 \cdot Y^3 = 27x^{6}Y^{3}$$ 3. **Rewrite the expression with these results:** $$\frac{4x^2}{27x^{6}Y^{3}}$$ 4. **Simplify the fraction by dividing coefficients and subtracting exponents of like bases:** $$\frac{4}{27} \cdot \frac{x^2}{x^{6}} \cdot \frac{1}{Y^{3}} = \frac{4}{27} \cdot x^{2-6} \cdot Y^{-3} = \frac{4}{27} x^{-4} Y^{-3}$$ 5. **Rewrite negative exponents as positive by moving factors to the denominator:** $$\frac{4}{27x^{4}Y^{3}}$$ **Final answer:** $$\boxed{\frac{4}{27x^{4}Y^{3}}}$$