\(\sqrt{500}+\sqrt{20}=\sqrt{100}⋅\sqrt{5}+\sqrt{4}⋅\sqrt{5}=10\sqrt{5}+2\sqrt{5}\text{.}\)

\(10\sqrt{5}+2\sqrt{5}=12\sqrt{5}\text{.}\)

\(\sqrt{20}=\sqrt{4}⋅\sqrt{5}=2\sqrt{5}\text{.}\)

\(-7\sqrt{175}=-7⋅\sqrt{25}⋅\sqrt{7}=-7⋅5⋅\sqrt{7}=-35\sqrt{7}\text{.}\)

\(6\sqrt{8}-7\sqrt{200}=6⋅\sqrt{4}⋅\sqrt{2}-7⋅\sqrt{100}⋅\sqrt{2}\text{.}\)

\(6⋅2⋅\sqrt{2}-7⋅10⋅\sqrt{2}=12\sqrt{2}-70\sqrt{2}=-58\sqrt{2}\text{.}\)

\(\sqrt{\frac{1}{4}}={\sqrt{1} \over \sqrt{4}}=\frac{1}{2}\text{.}\)

\({2 \over 7\sqrt{3}}={2 \over 7\sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={2\sqrt{3} \over 7⋅3}=\frac{2}{21}\sqrt{3}\text{.}\)

\(\sqrt{3\frac{21}{25}}=\sqrt{\frac{96}{25}}={\sqrt{96} \over \sqrt{25}}={\sqrt{96} \over 5}=\frac{1}{5}\sqrt{96}=\frac{1}{5}⋅4⋅\sqrt{6}=\frac{4}{5}\sqrt{6}\text{.}\)

\(\sqrt{\frac{25}{56}}={\sqrt{25} \over \sqrt{56}}={5 \over \sqrt{56}}⋅{\sqrt{56} \over \sqrt{56}}={5\sqrt{56} \over 56}=\frac{5}{56}\sqrt{56}=\frac{5}{56}⋅2⋅\sqrt{14}=\frac{5}{28}\sqrt{14}\text{.}\)

\(\sqrt{11\frac{1}{2}}=\sqrt{\frac{23}{2}}={\sqrt{23} \over \sqrt{2}}⋅{\sqrt{2} \over \sqrt{2}}={\sqrt{46} \over 2}=\frac{1}{2}\sqrt{46}\text{.}\)

\({36\sqrt{140} \over 4\sqrt{7}}={36 \over 4}⋅{\sqrt{140} \over \sqrt{7}}=9\sqrt{20}=9⋅\sqrt{4}⋅\sqrt{5}=9⋅2⋅\sqrt{5}=18\sqrt{5}\)

\(3\sqrt{14}⋅4\sqrt{7}=12\sqrt{98}=12⋅\sqrt{49}⋅\sqrt{2}=12⋅7⋅\sqrt{2}=84\sqrt{2}\)