Question
Rationalize the denominator and simplif \[ \sqrt[4]{\frac{3}{4}} \]
Ask by Davison Chen. in the United States
Mar 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form is \(\frac{\sqrt[4]{12}}{2}\).
Solution
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The Deep Dive
To rationalize the denominator of \( \sqrt[4]{\frac{3}{4}} \), we can rewrite this expression as: \[ \sqrt[4]{3} \div \sqrt[4]{4}. \] Since \( \sqrt[4]{4} = \sqrt{2} \), we have: \[ \frac{\sqrt[4]{3}}{\sqrt{2}}. \] Next, to rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{2} \): \[ \frac{\sqrt[4]{3} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt[4]{6}}{2}. \] Thus, the simplified and rationalized form of \( \sqrt[4]{\frac{3}{4}} \) is: \[ \frac{\sqrt[4]{6}}{2}. \]
