To solve the given expression \(\left(125 x^4 y^{\frac{3}{2}}\right)^{\frac{1}{3}} \div (x y)^{\frac{1}{2}}\):
1. Simplify the numerator \(\left(125 x^4 y^{\frac{3}{2}}\right)^{\frac{1}{3}}\):
Apply the power of a power rule \((a^m)^n = a^{mn}\):
[tex]\[
\left(125 x^4 y^{\frac{3}{2}}\right)^{\frac{1}{3}} = 125^{\frac{1}{3}} \cdot (x^4)^{\frac{1}{3}} \cdot \left(y^{\frac{3}{2}}\right)^{\frac{1}{3}}
\][/tex]
We know that \(125 = 5^3\), so:
[tex]\[
125^{\frac{1}{3}} = 5
\][/tex]
Simplify the exponents:
[tex]\[
(x^4)^{\frac{1}{3}} = x^{\frac{4}{3}}
\][/tex]
[tex]\[
\left(y^{\frac{3}{2}}\right)^{\frac{1}{3}} = y^{\frac{3}{2} \cdot \frac{1}{3}} = y^{\frac{1}{2}}
\][/tex]
Combining these, the numerator simplifies to:
[tex]\[
5 x^{\frac{4}{3}} y^{\frac{1}{2}}
\][/tex]
2. Simplify the denominator \((x y)^{\frac{1}{2}}\):
Apply the power of a product rule \((ab)^m = a^m b^m\):
[tex]\[
(x y)^{\frac{1}{2}} = x^{\frac{1}{2}} y^{\frac{1}{2}}
\][/tex]
3. Divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{5 x^{\frac{4}{3}} y^{\frac{1}{2}}}{x^{\frac{1}{2}} y^{\frac{1}{2}}}
\][/tex]
Use the quotient rule \(a^{m}/a^{n} = a^{m-n}\) to reduce the powers of \(x\) and \(y\):
[tex]\[
5 \cdot \frac{x^{\frac{4}{3}}}{x^{\frac{1}{2}}} \cdot \frac{y^{\frac{1}{2}}}{y^{\frac{1}{2}}}
\][/tex]
Simplify the \(y\)-terms:
[tex]\[
\frac{y^{\frac{1}{2}}}{y^{\frac{1}{2}}} = 1
\][/tex]
Simplify the \(x\)-terms:
[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]
Combines everything:
[tex]\[
5 \cdot x^{\frac{5}{6}}
\][/tex]
Therefore, the expression is equivalent to [tex]\(\boxed{5 x^{\frac{5}{6}}}\)[/tex].
