Answer
$\dfrac{3}{7\sqrt[3]{3}}$
Work Step by Step
Multiplying both the numerator and the denominator by a factor that will make the numerator a perfect power of the radical, the rationalized-numerator form of the given expression, $
\dfrac{\sqrt[3]{9}}{7}
,$ is
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[3]{9}}{7}\cdot\dfrac{\sqrt[3]{3}}{\sqrt[3]{3}}
\\\\=
\dfrac{\sqrt[3]{27}}{7\sqrt[3]{3}}
\\\\=
\dfrac{\sqrt[3]{(3)^3}}{7\sqrt[3]{3}}
\\\\=
\dfrac{3}{7\sqrt[3]{3}}
.\end{array}