Question
\( \frac{\sqrt[4]{y^{3}}}{\sqrt{y}} \) as a single radical? Explain.
Ask by Montgomery Carlson. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( \sqrt[4]{y} \).
Solution
Let’s rewrite the expression using exponent notation.
1. Write the numerator and denominator as exponents:
- The numerator: \( \sqrt[4]{y^3} = y^{\frac{3}{4}} \)
- The denominator: \( \sqrt{y} = y^{\frac{1}{2}} = y^{\frac{2}{4}} \)
2. Divide the exponents:
\[
\frac{y^{\frac{3}{4}}}{y^{\frac{2}{4}}} = y^{\frac{3}{4} - \frac{2}{4}} = y^{\frac{1}{4}}
\]
3. Express the result as a single radical:
\[
y^{\frac{1}{4}} = \sqrt[4]{y}
\]
Thus, the expression \( \frac{\sqrt[4]{y^{3}}}{\sqrt{y}} \) simplifies to \( \sqrt[4]{y} \).
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Simplify this solution The Deep Dive
To simplify \( \frac{\sqrt[4]{y^{3}}}{\sqrt{y}} \) as a single radical, start by expressing both radicals with exponents: - The fourth root of \( y^3 \) can be represented as \( y^{3/4} \). - The square root of \( y \) can be expressed as \( y^{1/2} \). Now, rewrite the expression: \[ \frac{y^{3/4}}{y^{1/2}} = y^{3/4 - 1/2} \] To combine the exponents, convert \( 1/2 \) to a fraction with the same denominator: \[ 1/2 = 2/4 \] Now, we have: \[ 3/4 - 2/4 = 1/4 \] Thus, \[ \frac{\sqrt[4]{y^{3}}}{\sqrt{y}} = y^{1/4} \] Finally, convert back to radical form: \[ \sqrt[4]{y} \] So, the expression simplifies to \( \sqrt[4]{y} \).
