Concept used:

In order to rationalize the numerator of an expression, we multiply in the numerator and denominator with such a term/expression such that we don't have any radicals left in the numerator upon simplifying.

Calcucation:

Let us rationalize the numerator as shown below:

\(\displaystyle\frac{{\sqrt[{{3}}]{{9}}}}{{7}}\) (Given expression)

\(\displaystyle\Rightarrow\frac{{\sqrt[{{3}}]{{9}}}}{{7}}\cdot\frac{{\sqrt[{{3}}]{{3}}}}{{\sqrt[{{3}}]{{3}}}}\ \text{(multiplying fraction by}\ \displaystyle{\sqrt[{{3}}]{{3}}})\)

\(\displaystyle\Rightarrow\frac{{{\sqrt[{{3}}]{{9}}}\cdot{\sqrt[{{3}}]{{3}}}}}{{{7}\cdot{\sqrt[{{3}}]{{3}}}}}\)

\(\displaystyle\Rightarrow\frac{{\sqrt[{{3}}]{{{9}\cdot{3}}}}}{{{7}{\sqrt[{{3}}]{{3}}}}}{\left({A}{p}{p}{l}{y}\ integer{\sqrt[{{n}}]{{a}}}\cdot{\sqrt[{{n}}]{{b}}}={\sqrt[{{n}}]{{{a}{b}}}}\right)}\)

\(\displaystyle\Rightarrow\frac{{\sqrt[{{3}}]{{27}}}}{{{7}{\sqrt[{{3}}]{{3}}}}}\)

\(\displaystyle\Rightarrow\frac{{\sqrt[{{3}}]{{{3}^{3}}}}}{{{7}{\sqrt[{{3}}]{{3}}}}}\)

\(\displaystyle\Rightarrow\frac{3}{{{7}{\sqrt[{{3}}]{{3}}}}}{\left({A}{p}{p}{l}{y}\ integer{\sqrt[{{n}}]{{{a}^{n}}}}={a}\right)}\)

Therefore, the rationalized form of the given expression would be \(\displaystyle\frac{3}{{{7}{\sqrt[{{3}}]{{3}}}}}.\)