What is \( \frac{\sqrt{25 x^{2} y^{2}}}{\sqrt{x y}} \) in simplest form? Assume \( x \geq 0 \) and \( y \geq 0 \)

To simplify \( \frac{\sqrt{25 x^{2} y^{2}}}{\sqrt{x y}} \), we start by simplifying the numerator and denominator separately. The numerator is: \[ \sqrt{25 x^2 y^2} = \sqrt{25} \cdot \sqrt{x^2} \cdot \sqrt{y^2} = 5xy \] The denominator is: \[ \sqrt{xy} \] Now we can rewrite our expression: \[ \frac{5xy}{\sqrt{xy}} \] Next, we simplify by expressing \( \sqrt{xy} \) as \( (xy)^{1/2} \): \[ \frac{5xy}{(xy)^{1/2}} = 5xy \cdot (xy)^{-1/2} = 5x^{1}\cdot y^{1} \cdot x^{-1/2}y^{-1/2} = 5x^{1-\frac{1}{2}}y^{1-\frac{1}{2}} = 5x^{\frac{1}{2}}y^{\frac{1}{2}} \] This can be rewritten as: \[ 5\sqrt{xy} \] Thus, the simplest form of \( \frac{\sqrt{25 x^{2} y^{2}}}{\sqrt{x y}} \) is: \[ \boxed{5\sqrt{xy}} \]