Answer
$\dfrac{9\sqrt[4]{4y^3}}{2y^2}$
Work Step by Step
Rationalizing the denominator of $
\dfrac{9y}{\sqrt[4]{4y^9}}
$ results to
\begin{array}{l}
\dfrac{9y}{\sqrt[4]{4y^9}}
\cdot
\dfrac{\sqrt[4]{4y^3}}{\sqrt[4]{4y^3}}
\\\\=
\dfrac{9y\sqrt[4]{4y^3}}{\sqrt[4]{16y^{12}}}
\\\\=
\dfrac{9y\sqrt[4]{4y^3}}{2y^3}
\\\\=
\dfrac{9\sqrt[4]{4y^3}}{2y^2}
.\end{array}