1. **State the problem:** Simplify the expression $$\left(\frac{2x^{3} y^{-3}}{3xy^{-2} \cdot 2y^{2}}\right)^{2}$$. 2. **Rewrite the denominator:** The denominator is a product: $$3xy^{-2} \cdot 2y^{2} = (3 \cdot 2)(x)(y^{-2} y^{2}) = 6x y^{-2+2} = 6x y^{0} = 6x$$. 3. **Rewrite the fraction inside the parentheses:** $$\frac{2x^{3} y^{-3}}{6x} = \frac{2}{6} \cdot \frac{x^{3}}{x} \cdot y^{-3} = \frac{1}{3} \cdot x^{3-1} \cdot y^{-3} = \frac{1}{3} x^{2} y^{-3}$$. 4. **Apply the square to the entire fraction:** $$\left(\frac{1}{3} x^{2} y^{-3}\right)^{2} = \left(\frac{1}{3}\right)^{2} \cdot (x^{2})^{2} \cdot (y^{-3})^{2} = \frac{1}{9} x^{4} y^{-6}$$. 5. **Final simplified expression:** $$\frac{x^{4}}{9 y^{6}}$$. **Explanation:** We simplified the denominator by multiplying terms and combining exponents. Then we simplified the fraction by dividing coefficients and subtracting exponents of like bases. Finally, we applied the power of 2 to each factor inside the parentheses, remembering to multiply exponents by 2.