(17) \( \frac{\frac{5 x+5 y}{x^{2} y^{2}}}{\frac{20}{x y}} \)

To simplify the expression: \[ \frac{\frac{5x + 5y}{x^{2} y^{2}}}{\frac{20}{x y}} \] **Step 1: Rewrite the Expression** Start by rewriting the complex fraction as a multiplication by the reciprocal: \[ \frac{5x + 5y}{x^{2} y^{2}} \div \frac{20}{x y} = \frac{5x + 5y}{x^{2} y^{2}} \times \frac{x y}{20} \] **Step 2: Factor Common Terms** Factor out the common term in the numerator: \[ 5x + 5y = 5(x + y) \] So the expression becomes: \[ \frac{5(x + y)}{x^{2} y^{2}} \times \frac{x y}{20} \] **Step 3: Multiply the Fractions** Multiply the numerators together and the denominators together: \[ \frac{5(x + y) \cdot x y}{x^{2} y^{2} \cdot 20} = \frac{5x y (x + y)}{20x^{2} y^{2}} \] **Step 4: Simplify Coefficients** Simplify the coefficients \( \frac{5}{20} \): \[ \frac{5}{20} = \frac{1}{4} \] **Step 5: Simplify Variables** Simplify the variable terms: \[ \frac{x y}{x^{2} y^{2}} = \frac{1}{x y} \] **Step 6: Combine Simplifications** Combine the simplified coefficients and variables: \[ \frac{1}{4} \times \frac{x + y}{x y} = \frac{x + y}{4x y} \] **Final Simplified Expression:** \[ \frac{x + y}{4x y} \]