\(\sqrt{500}+\sqrt{125}=\sqrt{100}⋅\sqrt{5}+\sqrt{25}⋅\sqrt{5}=10\sqrt{5}+5\sqrt{5}\text{.}\)

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

\(\sqrt{8}=\sqrt{4}⋅\sqrt{2}=2\sqrt{2}\text{.}\)

\(-7\sqrt{50}=-7⋅\sqrt{25}⋅\sqrt{2}=-7⋅5⋅\sqrt{2}=-35\sqrt{2}\text{.}\)

\(3\sqrt{12}+6\sqrt{75}=3⋅\sqrt{4}⋅\sqrt{3}+6⋅\sqrt{25}⋅\sqrt{3}\text{.}\)

\(3⋅2⋅\sqrt{3}+6⋅5⋅\sqrt{3}=6\sqrt{3}+30\sqrt{3}=36\sqrt{3}\text{.}\)

\(\sqrt{\frac{4}{49}}={\sqrt{4} \over \sqrt{49}}=\frac{2}{7}\text{.}\)

\({8 \over 3\sqrt{3}}={8 \over 3\sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={8\sqrt{3} \over 3⋅3}=\frac{8}{9}\sqrt{3}\text{.}\)

\(\sqrt{\frac{24}{49}}={\sqrt{24} \over \sqrt{49}}={\sqrt{24} \over 7}=\frac{1}{7}\sqrt{24}=\frac{1}{7}⋅2⋅\sqrt{6}=\frac{2}{7}\sqrt{6}\text{.}\)

\(\sqrt{\frac{1}{88}}={\sqrt{1} \over \sqrt{88}}={1 \over \sqrt{88}}⋅{\sqrt{88} \over \sqrt{88}}={\sqrt{88} \over 88}=\frac{1}{88}\sqrt{88}=\frac{1}{88}⋅2⋅\sqrt{22}=\frac{1}{44}\sqrt{22}\text{.}\)

\(\sqrt{4\frac{1}{3}}=\sqrt{\frac{13}{3}}={\sqrt{13} \over \sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={\sqrt{39} \over 3}=\frac{1}{3}\sqrt{39}\text{.}\)

\({54\sqrt{60} \over 9\sqrt{5}}={54 \over 9}⋅{\sqrt{60} \over \sqrt{5}}=6\sqrt{12}=6⋅\sqrt{4}⋅\sqrt{3}=6⋅2⋅\sqrt{3}=12\sqrt{3}\)

\(4\sqrt{10}⋅3\sqrt{2}=12\sqrt{20}=12⋅\sqrt{4}⋅\sqrt{5}=12⋅2⋅\sqrt{5}=24\sqrt{5}\)