\(\sqrt{63}=\sqrt{9}⋅\sqrt{7}=3\sqrt{7}\text{.}\)

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

\(\sqrt{\frac{9}{100}}={\sqrt{9} \over \sqrt{100}}=\frac{3}{10}\text{.}\)

\({54\sqrt{168} \over 9\sqrt{6}}={54 \over 9}⋅{\sqrt{168} \over \sqrt{6}}=6\sqrt{28}=6⋅\sqrt{4}⋅\sqrt{7}=6⋅2⋅\sqrt{7}=12\sqrt{7}\)

\(\sqrt{\frac{47}{49}}={\sqrt{47} \over \sqrt{49}}={\sqrt{47} \over 7}=\frac{1}{7}\sqrt{47}\text{.}\)

\(\sqrt{50}+\sqrt{200}=\sqrt{25}⋅\sqrt{2}+\sqrt{100}⋅\sqrt{2}=5\sqrt{2}+10\sqrt{2}\text{.}\)

\(5\sqrt{2}+10\sqrt{2}=15\sqrt{2}\text{.}\)

\(4\sqrt{27}+3\sqrt{300}=4⋅\sqrt{9}⋅\sqrt{3}+3⋅\sqrt{100}⋅\sqrt{3}\text{.}\)

\(4⋅3⋅\sqrt{3}+3⋅10⋅\sqrt{3}=12\sqrt{3}+30\sqrt{3}=42\sqrt{3}\text{.}\)

\({6 \over 5\sqrt{3}}={6 \over 5\sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={6\sqrt{3} \over 5⋅3}=\frac{2}{5}\sqrt{3}\text{.}\)

\(\sqrt{\frac{9}{76}}={\sqrt{9} \over \sqrt{76}}={3 \over \sqrt{76}}⋅{\sqrt{76} \over \sqrt{76}}={3\sqrt{76} \over 76}=\frac{3}{76}\sqrt{76}=\frac{3}{76}⋅2⋅\sqrt{19}=\frac{3}{38}\sqrt{19}\text{.}\)

\(\sqrt{10\frac{2}{3}}=\sqrt{\frac{32}{3}}={\sqrt{32} \over \sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={\sqrt{96} \over 3}=\frac{1}{3}\sqrt{96}=\frac{1}{3}⋅4⋅\sqrt{6}=1\frac{1}{3}\sqrt{6}\text{.}\)

\(2\sqrt{2}⋅4\sqrt{6}=8\sqrt{12}=8⋅\sqrt{4}⋅\sqrt{3}=8⋅2⋅\sqrt{3}=16\sqrt{3}\)