Answer
$\dfrac{\sqrt{2}}{3}$
Work Step by Step
Using the properties of radicals, the given expression, $
\dfrac{2\sqrt{3}}{3\sqrt{6}}
,$ evaluates to
\begin{array}{l}\require{cancel}
\dfrac{2}{3}\sqrt{\dfrac{3}{6}}
\\\\=
\dfrac{2}{3}\sqrt{\dfrac{1}{2}}
\\\\=
\dfrac{2}{3}\sqrt{\dfrac{1}{2}\cdot\dfrac{2}{2}}
\\\\=
\dfrac{2}{3}\sqrt{\dfrac{2}{4}}
\\\\=
\dfrac{2}{3}\cdot\dfrac{\sqrt{2}}{\sqrt{4}}
\\\\=
\dfrac{2}{3}\cdot\dfrac{\sqrt{2}}{2}
\\\\=
\dfrac{\cancel{2}}{3}\cdot\dfrac{\sqrt{2}}{\cancel{2}}
\\\\=
\dfrac{\sqrt{2}}{3}
.\end{array}
