Answer
$\dfrac{23\sqrt{5}}{30}$
Work Step by Step
Using the properties of radicals, the given expression, $
\dfrac{3}{\sqrt{5}}-\dfrac{2}{\sqrt{45}}+\dfrac{6}{\sqrt{80}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{3}{\sqrt{5}}-\dfrac{2}{\sqrt{9\cdot5}}+\dfrac{6}{\sqrt{16\cdot5}}
\\\\=
\dfrac{3}{\sqrt{5}}-\dfrac{2}{\sqrt{(3)^2\cdot5}}+\dfrac{6}{\sqrt{(4)^2\cdot5}}
\\\\=
\dfrac{3}{\sqrt{5}}-\dfrac{2}{3\sqrt{5}}+\dfrac{6}{4\sqrt{5}}
.\end{array}
Using the $LCD=
12\sqrt{5}
,$ the expression above simplifies to
\begin{array}{l}\require{cancel}
\dfrac{12(3)-4(2)+3(6)}{12\sqrt{5}}
\\\\=
\dfrac{36-8+18}{12\sqrt{5}}
\\\\=
\dfrac{46}{12\sqrt{5}}
\\\\=
\dfrac{\cancel{2}(23)}{\cancel{2}(6)\sqrt{5}}
\\\\=
\dfrac{23}{6\sqrt{5}}
.\end{array}
Rationalizing the denominator results to
\begin{array}{l}\require{cancel}
\dfrac{23}{6\sqrt{5}}\cdot\dfrac{\sqrt{5}}{\sqrt{5}}
\\\\=
\dfrac{23\sqrt{5}}{6(5)}
\\\\=
\dfrac{23\sqrt{5}}{30}
.\end{array}
