\(\sqrt{45}+\sqrt{20}=\sqrt{9}⋅\sqrt{5}+\sqrt{4}⋅\sqrt{5}=3\sqrt{5}+2\sqrt{5}\text{.}\)

\(3\sqrt{5}+2\sqrt{5}=5\sqrt{5}\text{.}\)

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

\(2\sqrt{112}=2⋅\sqrt{16}⋅\sqrt{7}=2⋅4⋅\sqrt{7}=8\sqrt{7}\text{.}\)

\(5\sqrt{50}+4\sqrt{18}=5⋅\sqrt{25}⋅\sqrt{2}+4⋅\sqrt{9}⋅\sqrt{2}\text{.}\)

\(5⋅5⋅\sqrt{2}+4⋅3⋅\sqrt{2}=25\sqrt{2}+12\sqrt{2}=37\sqrt{2}\text{.}\)

\(\sqrt{\frac{1}{100}}={\sqrt{1} \over \sqrt{100}}=\frac{1}{10}\text{.}\)

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

\(\sqrt{\frac{27}{100}}={\sqrt{27} \over \sqrt{100}}={\sqrt{27} \over 10}=\frac{1}{10}\sqrt{27}=\frac{1}{10}⋅3⋅\sqrt{3}=\frac{3}{10}\sqrt{3}\text{.}\)

\(\sqrt{\frac{25}{87}}={\sqrt{25} \over \sqrt{87}}={5 \over \sqrt{87}}⋅{\sqrt{87} \over \sqrt{87}}={5\sqrt{87} \over 87}=\frac{5}{87}\sqrt{87}\text{.}\)

\(\sqrt{3\frac{2}{3}}=\sqrt{\frac{11}{3}}={\sqrt{11} \over \sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={\sqrt{33} \over 3}=\frac{1}{3}\sqrt{33}\text{.}\)

\({56\sqrt{280} \over 8\sqrt{10}}={56 \over 8}⋅{\sqrt{280} \over \sqrt{10}}=7\sqrt{28}=7⋅\sqrt{4}⋅\sqrt{7}=7⋅2⋅\sqrt{7}=14\sqrt{7}\)

\(5\sqrt{2}⋅3\sqrt{10}=15\sqrt{20}=15⋅\sqrt{4}⋅\sqrt{5}=15⋅2⋅\sqrt{5}=30\sqrt{5}\)

\({5 \over 4-\sqrt{3}}={5 \over 4-\sqrt{3}}⋅{4+\sqrt{3} \over 4+\sqrt{3}}\)
\(\text{}={5(4-\sqrt{3}) \over 16-3}\)
\(\text{}=\frac{5}{13}(4-\sqrt{3})\)
\(\text{}=1\frac{7}{13}-\frac{5}{13}\sqrt{3}\)

\({\sqrt{3} \over \sqrt{5}+\sqrt{6}}={\sqrt{3} \over \sqrt{5}+\sqrt{6}}⋅{\sqrt{5}-\sqrt{6} \over \sqrt{5}-\sqrt{6}}\)
\(\text{}={\sqrt{3}(\sqrt{5}-\sqrt{6}) \over 5-6}\)
\(\text{}=-\sqrt{3}(\sqrt{5}-\sqrt{6})\)
\(\text{}=-\sqrt{15}+\sqrt{18}\)
\(\text{}=-\sqrt{15}+3\sqrt{2}\)