Answer
$-\dfrac{5\sqrt{3}}{6}$
Work Step by Step
Using the properties of radicals, the given expression, $
\dfrac{1}{3}\sqrt{12}-\dfrac{3}{2}\sqrt{48}+\dfrac{3}{4}\sqrt{108}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{1}{3}\sqrt{4\cdot3}-\dfrac{3}{2}\sqrt{16\cdot3}+\dfrac{3}{4}\sqrt{36\cdot3}
\\\\=
\dfrac{1}{3}\sqrt{(2)^2\cdot3}-\dfrac{3}{2}\sqrt{(4)^2\cdot3}+\dfrac{3}{4}\sqrt{(6)^2\cdot3}
\\\\=
\dfrac{1}{3}\cdot2\sqrt{3}-\dfrac{3}{2}\cdot4\sqrt{3}+\dfrac{3}{4}\cdot6\sqrt{3}
\\\\=
\dfrac{2}{3}\sqrt{3}-\dfrac{3}{\cancel{2}}\cdot\cancel{2}(2)\sqrt{3}+\dfrac{3}{\cancel{2}(2)}\cdot\cancel{2}(3)\sqrt{3}
\\\\=
\dfrac{2}{3}\sqrt{3}-6\sqrt{3}+\dfrac{9}{2}\sqrt{3}
\\\\=
\dfrac{4}{6}\sqrt{3}-\dfrac{36}{6}\sqrt{3}+\dfrac{27}{6}\sqrt{3}
\\\\=
-\dfrac{5}{6}\sqrt{3}
\\\\=
-\dfrac{5\sqrt{3}}{6}
.\end{array}
