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

To simplify the expression, start by expanding it step-by-step. The expression is: \[ \frac{5 x^{-2} y(3 x y)^{2}}{20 x y} \] First, expand the term \( (3xy)^2 \): \[ (3xy)^2 = 9x^2y^2 \] Now substitute this back into the expression: \[ \frac{5 x^{-2} y (9 x^2 y^2)}{20 x y} \] Next, multiply the numerator: \[ 5 x^{-2} y \cdot 9 x^2 y^2 = 45 x^{-2 + 2} y^{1 + 2} = 45 y^3 \] So now we have: \[ \frac{45 y^3}{20 x y} \] Next, simplify: \[ \frac{45 y^3}{20 x y} = \frac{45}{20} \cdot \frac{y^3}{y} \] This reduces to: \[ \frac{45}{20} \cdot y^{3-1} = \frac{45}{20} \cdot y^2 \] Now simplify \(\frac{45}{20}\): \[ \frac{45}{20} = \frac{9}{4} \text{ (dividing both numerator and denominator by 5)} \] Thus, the final simplified expression is: \[ \frac{9 y^2}{4 x} \] So the complete solution is: \[ \frac{9 y^2}{4 x} \]