Use the quotient rule to simplify, Assume all variables represent non-negative values. \( \sqrt{\frac{32 x^{3}}{25 y^{2}}} \) \( \sqrt{\frac{32 x^{3}}{25 y^{2}}}=\square \) (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
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To simplify \( \sqrt{\frac{32 x^{3}}{25 y^{2}}} \), we can apply the quotient rule for square roots, which states that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Therefore, we can rewrite the expression as: \[ \frac{\sqrt{32 x^{3}}}{\sqrt{25 y^{2}}}. \] Next, we simplify the numerator and denominator separately. The denominator simplifies to: \[ \sqrt{25 y^{2}} = 5y. \] Now for the numerator, we have: \[ \sqrt{32 x^{3}} = \sqrt{16 \cdot 2 \cdot x^{2} \cdot x} = \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{x^{2}} \cdot \sqrt{x} = 4\sqrt{2} \cdot x \cdot \sqrt{x} = 4x\sqrt{2x}. \] Putting it all together gives us: \[ \frac{4x \sqrt{2x}}{5y}. \] Thus, the simplified expression is: \[ \sqrt{\frac{32 x^{3}}{25 y^{2}}} = \frac{4x \sqrt{2x}}{5y}. \]