4. \( \frac{5 x^{-2} y(3 x y)^{2}}{20 x y} \)

To simplify the expression: \[ \frac{5 x^{-2} y (3 x y)^{2}}{20 x y} \] **Step-by-Step Simplification:** 1. **Expand \((3 x y)^2\):** \[ (3 x y)^2 = 3^2 \cdot x^2 \cdot y^2 = 9 x^2 y^2 \] 2. **Substitute back into the original expression:** \[ \frac{5 x^{-2} y \cdot 9 x^2 y^2}{20 x y} = \frac{45 x^{-2} \cdot x^2 \cdot y \cdot y^2}{20 x y} \] 3. **Simplify the \(x\) terms:** \[ x^{-2} \cdot x^2 = x^{(-2 + 2)} = x^0 = 1 \] 4. **Simplify the \(y\) terms:** \[ y \cdot y^2 = y^{1+2} = y^3 \] 5. **Now, the expression becomes:** \[ \frac{45 \cdot y^3}{20 x y} = \frac{45 y^3}{20 x y} \] 6. **Simplify the coefficients and the remaining \(y\) terms:** - Coefficients: \( \frac{45}{20} = \frac{9}{4} \) - \( y^3 / y = y^{3-1} = y^2 \) 7. **Final Simplified Expression:** \[ \frac{9 y^2}{4 x} \] **Answer:** \[ \frac{9\, y^{2}}{4\, x} \]