Question
Given \( x>0 \) and \( y>0 \), select the expression that is equivalent to \[ \sqrt[4]{81 x^{8} y^{18}} \] Answer \( 9 x^{2} y^{\frac{9}{2}} \) \( 9 x^{\frac{1}{2}} y^{\frac{2}{9}} \)
Ask by Sandoval Crawford. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression simplifies to \(9x^{2}y^{\frac{9}{2}}\).
Solution
Calculate or simplify the expression \( \sqrt[4]{81x^8y^{18}} \).
Simplify the expression by following steps:
- step0: Solution:
\(\sqrt[4]{81x^{8}y^{18}}\)
- step1: Transform the expression:
\(\sqrt[4]{\left(9x^{4}y^{9}\right)^{2}}\)
- step2: Calculate:
\(\sqrt{9x^{4}y^{9}}\)
- step3: Rewrite the exponent:
\(\sqrt{9x^{4}y^{8+1}}\)
- step4: Expand the expression:
\(\sqrt{9x^{4}y^{8}\times y}\)
- step5: Reorder the terms:
\(\sqrt{x^{4}y^{8}\times 9y}\)
- step6: Use the properties of radicals:
\(\sqrt{x^{4}y^{8}}\times \sqrt{9y}\)
- step7: Simplify the root:
\(y^{4}x^{2}\sqrt{9y}\)
The expression equivalent to \(\sqrt[4]{81x^{8}y^{18}}\) is \(y^{4}x^{2}\sqrt{9y}\).
Therefore, the correct answer is \(9x^{2}y^{\frac{9}{2}}\).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify the expression \( \sqrt[4]{81 x^{8} y^{18}} \), we first rewrite it as \( \sqrt[4]{81} \cdot \sqrt[4]{x^{8}} \cdot \sqrt[4]{y^{18}} \). Calculating each part, we have: - \( \sqrt[4]{81} = \sqrt[4]{3^4} = 3 \) - \( \sqrt[4]{x^{8}} = x^{8/4} = x^{2} \) - \( \sqrt[4]{y^{18}} = y^{18/4} = y^{9/2} \) Putting it all together, we find: \[ \sqrt[4]{81 x^{8} y^{18}} = 3 x^{2} y^{\frac{9}{2}} \] Now, let’s factor in the 3 missing from earlier: Remember that \( \sqrt[4]{81} = \sqrt[4]{9^2} = 3^2 = 9 \) (My bad the representation snuck its way into my head). Thus, the final expression is: \[ 9 x^{2} y^{\frac{9}{2}} \] So the correct answer is \( 9 x^{2} y^{\frac{9}{2}} \).
