Answer
$2\sqrt{5}$
Work Step by Step
Using the properties of radicals, the given expression, $
\dfrac{1}{2}\sqrt{20}+\dfrac{2}{3}\sqrt{45}-\dfrac{1}{4}\sqrt{80}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{1}{2}\sqrt{4\cdot5}+\dfrac{2}{3}\sqrt{9\cdot5}-\dfrac{1}{4}\sqrt{16\cdot5}
\\\\=
\dfrac{1}{2}\sqrt{(2)^2\cdot5}+\dfrac{2}{3}\sqrt{(3)^2\cdot5}-\dfrac{1}{4}\sqrt{(4)^2\cdot5}
\\\\=
\dfrac{1}{2}\cdot2\sqrt{5}+\dfrac{2}{3}\cdot3\sqrt{5}-\dfrac{1}{4}\cdot4\sqrt{5}
\\\\=
\sqrt{5}+2\sqrt{5}-\sqrt{5}
\\\\=
2\sqrt{5}
.\end{array}
