Answer
$\dfrac{\sqrt[3]{b^2}}{b}$
Work Step by Step
Multiplying both the numerator and the denominator by a factor that will make the denominator a perfect power of the radical, the rationalized-denominator form of the given expression, $
\dfrac{\sqrt[3]{ab}}{\sqrt[3]{ab^2}}
,$ is
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[3]{ab}}{\sqrt[3]{ab^2}}\cdot\dfrac{\sqrt[3]{a^2b}}{\sqrt[3]{a^2b}}
\\\\=
\dfrac{\sqrt[3]{a^3b^2}}{\sqrt[3]{a^3b^3}}
\\\\=
\dfrac{\sqrt[3]{(a)^3\cdot b^2}}{\sqrt[3]{(ab)^3}}
\\\\=
\dfrac{a\sqrt[3]{b^2}}{ab}
\\\\=
\dfrac{\cancel{a}\sqrt[3]{b^2}}{\cancel{a}\cdot b}
\\\\=
\dfrac{\sqrt[3]{b^2}}{b}
.\end{array}