Answer
$\displaystyle \frac{7\sqrt{6}}{6}$
Work Step by Step
$\displaystyle \sqrt{\frac{1}{6}}=\frac{\sqrt{1}}{\sqrt{6}}=\frac{1}{\sqrt{6}}\color{red}{ \cdot\frac{ \sqrt{6}}{\sqrt{6}} }= \frac{\sqrt{6}}{6}$
$\displaystyle \sqrt{\frac{2}{3}}=\sqrt{\frac{4}{6}}=\frac{\sqrt{4}}{\sqrt{6}}=\frac{2}{\sqrt{6}}\color{red}{ \cdot\frac{ \sqrt{6}}{\sqrt{6}} }=\frac{2\sqrt{6}}{6}$
$\displaystyle \sqrt{6}-\sqrt{\frac{1}{6}}+\sqrt{\frac{2}{3}}= \frac{6\sqrt{6}}{6}- \frac{\sqrt{6}}{6}+ \frac{2\sqrt{6}}{6}$
$= \displaystyle \frac{7\sqrt{6}}{6}$
