Answer
$\dfrac{\sqrt[]{3}}{3}$
Work Step by Step
Multiplying by an expression equal to $1$ such that the denominator becomes a perfect power of the index, the given expression, $
\sqrt[4]{\dfrac{1}{9}}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[4]{\dfrac{1}{9}\cdot\dfrac{9}{9}}
\\=
\sqrt[4]{\dfrac{9}{81}}
\\=
\sqrt[4]{\dfrac{9}{3^4}}
\\=
\sqrt[4]{\dfrac{1}{3^4}\cdot9}
\\=
\sqrt[4]{\left(\dfrac{1}{3}\right)^4\cdot9}
\\=
\dfrac{1}{3}\sqrt[4]{9}
\\=
\dfrac{1}{3}\sqrt[4]{3^2}
\\=
\dfrac{1}{3}\sqrt[\frac{4}{2}]{3^{\frac{2}{2}}}
\\=
\dfrac{1}{3}\sqrt[2]{3^{1}}
\\=
\dfrac{1}{3}\sqrt[]{3}
\\=
\dfrac{\sqrt[]{3}}{3}
.\end{array}
