\(\sqrt{125}+\sqrt{45}=\sqrt{25}⋅\sqrt{5}+\sqrt{9}⋅\sqrt{5}=5\sqrt{5}+3\sqrt{5}\text{.}\)

\(5\sqrt{5}+3\sqrt{5}=8\sqrt{5}\text{.}\)

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

\(-4\sqrt{48}=-4⋅\sqrt{16}⋅\sqrt{3}=-4⋅4⋅\sqrt{3}=-16\sqrt{3}\text{.}\)

\(3\sqrt{50}+4\sqrt{18}=3⋅\sqrt{25}⋅\sqrt{2}+4⋅\sqrt{9}⋅\sqrt{2}\text{.}\)

\(3⋅5⋅\sqrt{2}+4⋅3⋅\sqrt{2}=15\sqrt{2}+12\sqrt{2}=27\sqrt{2}\text{.}\)

\(\sqrt{3\frac{1}{16}}=\sqrt{\frac{49}{16}}={\sqrt{49} \over \sqrt{16}}=\frac{7}{4}=1\frac{3}{4}\text{.}\)

\({2 \over 9\sqrt{2}}={2 \over 9\sqrt{2}}⋅{\sqrt{2} \over \sqrt{2}}={2\sqrt{2} \over 9⋅2}=\frac{1}{9}\sqrt{2}\text{.}\)

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

\(\sqrt{1\frac{1}{24}}=\sqrt{\frac{25}{24}}={\sqrt{25} \over \sqrt{24}}={5 \over \sqrt{24}}⋅{\sqrt{24} \over \sqrt{24}}={5\sqrt{24} \over 24}=\frac{5}{24}\sqrt{24}=\frac{5}{24}⋅2⋅\sqrt{6}=\frac{5}{12}\sqrt{6}\text{.}\)

\(\sqrt{\frac{2}{35}}={\sqrt{2} \over \sqrt{35}}⋅{\sqrt{35} \over \sqrt{35}}={\sqrt{70} \over 35}=\frac{1}{35}\sqrt{70}\text{.}\)

\({32\sqrt{168} \over 8\sqrt{7}}={32 \over 8}⋅{\sqrt{168} \over \sqrt{7}}=4\sqrt{24}=4⋅\sqrt{4}⋅\sqrt{6}=4⋅2⋅\sqrt{6}=8\sqrt{6}\)

\(5\sqrt{3}⋅3\sqrt{15}=15\sqrt{45}=15⋅\sqrt{9}⋅\sqrt{5}=15⋅3⋅\sqrt{5}=45\sqrt{5}\)

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

\({4\sqrt{6} \over \sqrt{5}-\sqrt{3}}={4\sqrt{6} \over \sqrt{5}-\sqrt{3}}⋅{\sqrt{5}+\sqrt{3} \over \sqrt{5}+\sqrt{3}}\)
\(\text{}={4\sqrt{6}(\sqrt{5}+\sqrt{3}) \over 5-3}\)
\(\text{}=2\sqrt{6}(\sqrt{5}+\sqrt{3})\)
\(\text{}=2\sqrt{30}+2\sqrt{18}\)
\(\text{}=2\sqrt{30}+6\sqrt{2}\)