\(\sqrt{500}+\sqrt{80}=\sqrt{100}⋅\sqrt{5}+\sqrt{16}⋅\sqrt{5}=10\sqrt{5}+4\sqrt{5}\text{.}\)

\(10\sqrt{5}+4\sqrt{5}=14\sqrt{5}\text{.}\)

\(\sqrt{500}=\sqrt{100}⋅\sqrt{5}=10\sqrt{5}\text{.}\)

\(-6\sqrt{50}=-6⋅\sqrt{25}⋅\sqrt{2}=-6⋅5⋅\sqrt{2}=-30\sqrt{2}\text{.}\)

\(3\sqrt{125}+7\sqrt{80}=3⋅\sqrt{25}⋅\sqrt{5}+7⋅\sqrt{16}⋅\sqrt{5}\text{.}\)

\(3⋅5⋅\sqrt{5}+7⋅4⋅\sqrt{5}=15\sqrt{5}+28\sqrt{5}=43\sqrt{5}\text{.}\)

\(\sqrt{\frac{1}{64}}={\sqrt{1} \over \sqrt{64}}=\frac{1}{8}\text{.}\)

\({7 \over 4\sqrt{3}}={7 \over 4\sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={7\sqrt{3} \over 4⋅3}=\frac{7}{12}\sqrt{3}\text{.}\)

\(\sqrt{\frac{79}{81}}={\sqrt{79} \over \sqrt{81}}={\sqrt{79} \over 9}=\frac{1}{9}\sqrt{79}\text{.}\)

\(\sqrt{\frac{9}{50}}={\sqrt{9} \over \sqrt{50}}={3 \over \sqrt{50}}⋅{\sqrt{50} \over \sqrt{50}}={3\sqrt{50} \over 50}=\frac{3}{50}\sqrt{50}=\frac{3}{50}⋅5⋅\sqrt{2}=\frac{3}{10}\sqrt{2}\text{.}\)

\(\sqrt{2\frac{4}{5}}=\sqrt{\frac{14}{5}}={\sqrt{14} \over \sqrt{5}}⋅{\sqrt{5} \over \sqrt{5}}={\sqrt{70} \over 5}=\frac{1}{5}\sqrt{70}\text{.}\)

\({45\sqrt{144} \over 9\sqrt{6}}={45 \over 9}⋅{\sqrt{144} \over \sqrt{6}}=5\sqrt{24}=5⋅\sqrt{4}⋅\sqrt{6}=5⋅2⋅\sqrt{6}=10\sqrt{6}\)

\(4\sqrt{14}⋅2\sqrt{6}=8\sqrt{84}=8⋅\sqrt{4}⋅\sqrt{21}=8⋅2⋅\sqrt{21}=16\sqrt{21}\)

\({-3 \over 1-\sqrt{5}}={-3 \over 1-\sqrt{5}}⋅{1+\sqrt{5} \over 1+\sqrt{5}}\)
\(\text{}={-3(1-\sqrt{5}) \over 1-5}\)
\(\text{}=\frac{3}{4}(1-\sqrt{5})\)
\(\text{}=\frac{3}{4}-\frac{3}{4}\sqrt{5}\)

\({\sqrt{6} \over \sqrt{5}-\sqrt{3}}={\sqrt{6} \over \sqrt{5}-\sqrt{3}}⋅{\sqrt{5}+\sqrt{3} \over \sqrt{5}+\sqrt{3}}\)
\(\text{}={\sqrt{6}(\sqrt{5}+\sqrt{3}) \over 5-3}\)
\(\text{}=\frac{1}{2}\sqrt{6}(\sqrt{5}+\sqrt{3})\)
\(\text{}=\frac{1}{2}\sqrt{30}+\frac{1}{2}\sqrt{18}\)
\(\text{}=\frac{1}{2}\sqrt{30}+1\frac{1}{2}\sqrt{2}\)