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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.)

Ask by Campbell Daniels. in the United States
Jan 30,2025

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Tutor-Verified Answer

Answer

\( \frac{4x\sqrt{2x}}{5y} \)

Solution

Calculate or simplify the expression \( \sqrt{\frac{32x^3}{25y^2}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{\frac{32x^{3}}{25y^{2}}}\) - step1: Use the properties of radicals: \(\frac{\sqrt{32x^{3}}}{\sqrt{25y^{2}}}\) - step2: Simplify the expression: \(\frac{4x\sqrt{2x}}{5y}\) The simplified form of \( \sqrt{\frac{32x^{3}}{25y^{2}}} \) is \( \frac{4x\sqrt{2x}}{5y} \).

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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: \[ \sqrt{\frac{32 x^{3}}{25 y^{2}}} = \frac{4x \sqrt{2x}}{5y}. \]

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