To simplify the expression \(\sqrt{\frac{5ab^3}{3}}\), we can break it down as follows:
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Split the square root into two parts: \[ \sqrt{\frac{5ab^3}{3}} = \frac{\sqrt{5ab^3}}{\sqrt{3}} \]
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Now, simplify the square root of the numerator: \[ \sqrt{5ab^3} = \sqrt{5} \cdot \sqrt{a} \cdot \sqrt{b^3} \]
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Since \(b^3 = b^2 \cdot b\), we can simplify \(\sqrt{b^3}\): \[ \sqrt{b^3} = \sqrt{b^2 \cdot b} = \sqrt{b^2} \cdot \sqrt{b} = b \cdot \sqrt{b} \]
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Combine everything together: \[ \sqrt{5ab^3} = \sqrt{5} \cdot \sqrt{a} \cdot b \cdot \sqrt{b} = b\sqrt{5a} \cdot \sqrt{b} \] Therefore: \[ \sqrt{5ab^3} = b\sqrt{5ab} \]
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Now, put it all back into the expression: \[ \frac{\sqrt{5ab^3}}{\sqrt{3}} = \frac{b\sqrt{5ab}}{\sqrt{3}} \]
The final simplified expression is: \[ \frac{b\sqrt{5ab}}{\sqrt{3}} \]