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\(\sqrt{\frac{9}{25}} = {\sqrt{9} \over \sqrt{25}} = \frac{3}{5} \text{.}\)

\(\sqrt{\frac{89}{100}} = {\sqrt{89} \over \sqrt{100}} = {\sqrt{89} \over 10} = \frac{1}{10} \sqrt{89} \text{.}\)

\(\sqrt{2\frac{9}{20}} = \sqrt{\frac{49}{20}} = {\sqrt{49} \over \sqrt{20}} = {7 \over \sqrt{20}} ⋅ {\sqrt{20} \over \sqrt{20}} = {7 \sqrt{20} \over 20} = \frac{7}{20} \sqrt{20} = \frac{7}{20} ⋅ 2 ⋅ \sqrt{5} = \frac{7}{10} \sqrt{5} \text{.}\)

\(\sqrt{37\frac{1}{2}} = \sqrt{\frac{75}{2}} = {\sqrt{75} \over \sqrt{2}} ⋅ {\sqrt{2} \over \sqrt{2}} = {\sqrt{150} \over 2} = \frac{1}{2} \sqrt{150} = \frac{1}{2} ⋅ 5 ⋅ \sqrt{6} = 2\frac{1}{2} \sqrt{6} \text{.}\)

\({8 \sqrt{240} \over 2 \sqrt{10}} = {8 \over 2} ⋅ {\sqrt{240} \over \sqrt{10}} = 4 \sqrt{24} = 4 ⋅ \sqrt{4} ⋅ \sqrt{6} = 4 ⋅ 2 ⋅ \sqrt{6} = 8 \sqrt{6}\)

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

\(3 \sqrt{75} = 3 ⋅ \sqrt{25} ⋅ \sqrt{3} = 3 ⋅ 5 ⋅ \sqrt{3} = 15 \sqrt{3} \text{.}\)

\(\sqrt{32} + \sqrt{50} = \sqrt{16} ⋅ \sqrt{2} + \sqrt{25} ⋅ \sqrt{2} = 4 \sqrt{2} + 5 \sqrt{2} \text{.}\)

\(4 \sqrt{2} + 5 \sqrt{2} = 9 \sqrt{2} \text{.}\)

\(5 \sqrt{50} + 3 \sqrt{18} = 5 ⋅ \sqrt{25} ⋅ \sqrt{2} + 3 ⋅ \sqrt{9} ⋅ \sqrt{2} \text{.}\)

\(5 ⋅ 5 ⋅ \sqrt{2} + 3 ⋅ 3 ⋅ \sqrt{2} = 25 \sqrt{2} + 9 \sqrt{2} = 34 \sqrt{2} \text{.}\)

\({3 \over 4 + \sqrt{6}} = {3 \over 4 + \sqrt{6}} ⋅ {4 - \sqrt{6} \over 4 - \sqrt{6}}\)
\(\text{} = {3 (4 + \sqrt{6}) \over 16 - 6}\)
\(\text{} = \frac{3}{10} (4 + \sqrt{6})\)
\(\text{} = 1\frac{1}{5} + \frac{3}{10} \sqrt{6}\)

\({4 \sqrt{2} \over \sqrt{6} - \sqrt{5}} = {4 \sqrt{2} \over \sqrt{6} - \sqrt{5}} ⋅ {\sqrt{6} + \sqrt{5} \over \sqrt{6} + \sqrt{5}}\)
\(\text{} = {4 \sqrt{2} (\sqrt{6} + \sqrt{5}) \over 6 - 5}\)
\(\text{} = 4 \sqrt{2} (\sqrt{6} + \sqrt{5})\)
\(\text{} = 4 \sqrt{12} + 4 \sqrt{10}\)
\(\text{} = 8 \sqrt{3} + 4 \sqrt{10}\)

\(4 \sqrt{6} ⋅ 3 \sqrt{2} = 12 \sqrt{12} = 12 ⋅ \sqrt{4} ⋅ \sqrt{3} = 12 ⋅ 2 ⋅ \sqrt{3} = 24 \sqrt{3}\)

\({7 \over 2 \sqrt{3}} = {7 \over 2 \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {7 \sqrt{3} \over 2 ⋅ 3} = 1\frac{1}{6} \sqrt{3} \text{.}\)