Use the quotient rule to simplify. Assume that all variables represent positive real numbers. \( \sqrt{\frac{x^{2} y}{225}} \) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

To simplify \( \sqrt{\frac{x^{2} y}{225}} \) using the rules of radicals, we start by applying the quotient rule for square roots, which states that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). So, we rewrite the expression: \[ \sqrt{\frac{x^{2} y}{225}} = \frac{\sqrt{x^{2} y}}{\sqrt{225}}. \] Next, we simplify \( \sqrt{225} \), which is \( 15 \). Now, simplifying \( \sqrt{x^{2} y} \) gives us \( \sqrt{x^{2}} \cdot \sqrt{y} = x\sqrt{y} \). Putting it all together, we have: \[ \frac{\sqrt{x^{2} y}}{\sqrt{225}} = \frac{x\sqrt{y}}{15}. \] Thus, the simplified expression is: \[ \frac{x\sqrt{y}}{15}. \]