To determine which expression is equivalent to [tex]\(\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}\)[/tex], we need to simplify the given expression step by step.
1. Start with the given expression:
[tex]\[
\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}
\][/tex]
2. Simplify the fraction inside the radical:
[tex]\[
\frac{24 x^6 y}{128 x^4 y^5} = \frac{24}{128} \cdot \frac{x^6}{x^4} \cdot \frac{y}{y^5}
\][/tex]
3. Simplify each term separately:
- Simplify the numerical fraction:
[tex]\[
\frac{24}{128} = \frac{24 \div 8}{128 \div 8} = \frac{3}{16}
\][/tex]
- Simplify the powers of [tex]\(x\)[/tex]:
[tex]\[
\frac{x^6}{x^4} = x^{6-4} = x^2
\][/tex]
- Simplify the powers of [tex]\(y\)[/tex]:
[tex]\[
\frac{y}{y^5} = y^{1-5} = y^{-4}
\][/tex]
4. Combine all simplified terms:
[tex]\[
\frac{3}{16} \cdot x^2 \cdot y^{-4} = \frac{3 x^2}{16 y^4}
\][/tex]
5. Now we have:
[tex]\[
\sqrt[4]{\frac{3 x^2}{16 y^4}}
\][/tex]
6. Separately apply the fourth root to the numerator and the denominator:
[tex]\[
\sqrt[4]{\frac{3 x^2}{16 y^4}} = \frac{\sqrt[4]{3 x^2}}{\sqrt[4]{16 y^4}}
\][/tex]
7. Compute the fourth root of the numerator and denominator:
- The denominator:
[tex]\[
\sqrt[4]{16 y^4} = \sqrt[4]{16} \cdot \sqrt[4]{y^4} = 2 \cdot y
\][/tex]
- The numerator:
[tex]\[
\sqrt[4]{3 x^2} = 3^{1/4} \cdot (x^2)^{1/4} = 3^{1/4} \cdot x^{2/4} = 3^{1/4} \cdot x^{1/2}
\][/tex]
8. Combine these results:
[tex]\[
\frac{\sqrt[4]{3 x^2}}{\sqrt[4]{16 y^4}} = \frac{3^{1/4} \cdot x^{1/2}}{2 y}
\][/tex]
9. Recognize that [tex]\(x^{1/2}\)[/tex] is the same as [tex]\(\sqrt{x}\)[/tex]:
[tex]\[
\frac{3^{1/4} \sqrt{x}}{2 y}
\][/tex]
Therefore, the correct equivalent expression is:
[tex]\[
\frac{\sqrt[4]{3 x^2}}{2 y}
\][/tex]
So the corresponding option is:
[tex]\(\boxed{\frac{\sqrt[4]{3 x^2}}{2 y}}\)[/tex] which matches option [tex]\(4\)[/tex].
