To determine the equivalent expression for [tex]\(\left(125 x^4 y^{\frac{2}{2}}\right)^{\frac{1}{3}} \div (x y)^{\frac{1}{2}}\)[/tex], we follow a step-by-step simplification:
1. Start with the given expression:
[tex]\[
\left(125 x^4 y^{\frac{2}{2}}\right)^{\frac{1}{3}} \div (x y)^{\frac{1}{2}}
\][/tex]
2. Simplify the exponent of [tex]\(y\)[/tex] in the numerator:
[tex]\[
y^{\frac{2}{2}} = y
\][/tex]
So the expression becomes:
[tex]\[
\left(125 x^4 y\right)^{\frac{1}{3}} \div (x y)^{\frac{1}{2}}
\][/tex]
3. Distribute the exponent in the numerator:
[tex]\[
\left(125\right)^{\frac{1}{3}} \left(x^4\right)^{\frac{1}{3}} \left(y\right)^{\frac{1}{3}}
\][/tex]
4. Calculate the individual components:
[tex]\[
125^{\frac{1}{3}} = 5
\][/tex]
[tex]\[
(x^4)^{\frac{1}{3}} = x^{\frac{4}{3}}
\][/tex]
[tex]\[
y^{\frac{1}{3}}
\][/tex]
So, the numerator becomes:
[tex]\[
5 x^{\frac{4}{3}} y^{\frac{1}{3}}
\][/tex]
5. Simplify the denominator:
[tex]\[
(x y)^{\frac{1}{2}} = x^{\frac{1}{2}} y^{\frac{1}{2}}
\][/tex]
6. Combine the numerator and the denominator:
[tex]\[
\frac{5 x^{\frac{4}{3}} y^{\frac{1}{3}}}{x^{\frac{1}{2}} y^{\frac{1}{2}}}
\][/tex]
7. Subtract the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the numerator and the denominator:
[tex]\[
x^{\frac{4}{3} - \frac{1}{2}} = x^{\frac{4}{3} - \frac{1}{2}} = x^{\frac{8}{6} - \frac{3}{6}} = x^{\frac{5}{6}}
\][/tex]
[tex]\[
y^{\frac{1}{3} - \frac{1}{2}} = y^{\frac{1}{3} - \frac{1}{2}} = y^{\frac{2}{6} - \frac{3}{6}} = y^{-\frac{1}{6}}
\][/tex]
8. Hence, the simplified expression is:
[tex]\[
5 x^{\frac{5}{6}} y^{-\frac{1}{6}}
\][/tex]
Since [tex]\(y^{-\frac{1}{6}}\)[/tex] means [tex]\( \frac{1}{y^{\frac{1}{6}}}\)[/tex], the final result simplifies the expression to:
[tex]\[
5 x^{\frac{5}{6}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{5 x^{\frac{5}{6}}}
\][/tex]
