\(\sqrt{20} + \sqrt{500} = \sqrt{4} ⋅ \sqrt{5} + \sqrt{100} ⋅ \sqrt{5} = 2 \sqrt{5} + 10 \sqrt{5} \text{.}\)

\(2 \sqrt{5} + 10 \sqrt{5} = 12 \sqrt{5} \text{.}\)

\(\sqrt{80} = \sqrt{16} ⋅ \sqrt{5} = 4 \sqrt{5} \text{.}\)

\(5 \sqrt{125} = 5 ⋅ \sqrt{25} ⋅ \sqrt{5} = 5 ⋅ 5 ⋅ \sqrt{5} = 25 \sqrt{5} \text{.}\)

\(6 \sqrt{32} - 7 \sqrt{8} = 6 ⋅ \sqrt{16} ⋅ \sqrt{2} - 7 ⋅ \sqrt{4} ⋅ \sqrt{2} \text{.}\)

\(6 ⋅ 4 ⋅ \sqrt{2} - 7 ⋅ 2 ⋅ \sqrt{2} = 24 \sqrt{2} - 14 \sqrt{2} = 10 \sqrt{2} \text{.}\)

\(\sqrt{\frac{25}{49}} = {\sqrt{25} \over \sqrt{49}} = \frac{5}{7} \text{.}\)

\({5 \over 4 \sqrt{7}} = {5 \over 4 \sqrt{7}} ⋅ {\sqrt{7} \over \sqrt{7}} = {5 \sqrt{7} \over 4 ⋅ 7} = \frac{5}{28} \sqrt{7} \text{.}\)

\(\sqrt{\frac{17}{64}} = {\sqrt{17} \over \sqrt{64}} = {\sqrt{17} \over 8} = \frac{1}{8} \sqrt{17} \text{.}\)

\(\sqrt{1\frac{1}{35}} = \sqrt{\frac{36}{35}} = {\sqrt{36} \over \sqrt{35}} = {6 \over \sqrt{35}} ⋅ {\sqrt{35} \over \sqrt{35}} = {6 \sqrt{35} \over 35} = \frac{6}{35} \sqrt{35} \text{.}\)

\(\sqrt{1\frac{3}{8}} = \sqrt{\frac{11}{8}} = {\sqrt{11} \over \sqrt{8}} ⋅ {\sqrt{8} \over \sqrt{8}} = {\sqrt{88} \over 8} = \frac{1}{8} \sqrt{88} = \frac{1}{8} ⋅ 2 ⋅ \sqrt{22} = \frac{1}{4} \sqrt{22} \text{.}\)

\({9 \sqrt{168} \over \sqrt{6}} = 9 ⋅ {\sqrt{168} \over \sqrt{6}} = 9 \sqrt{28} = 9 ⋅ \sqrt{4} ⋅ \sqrt{7} = 9 ⋅ 2 ⋅ \sqrt{7} = 18 \sqrt{7}\)

\(2 \sqrt{2} ⋅ 4 \sqrt{10} = 8 \sqrt{20} = 8 ⋅ \sqrt{4} ⋅ \sqrt{5} = 8 ⋅ 2 ⋅ \sqrt{5} = 16 \sqrt{5}\)

\({2 \over 4 - \sqrt{6}} = {2 \over 4 - \sqrt{6}} ⋅ {4 + \sqrt{6} \over 4 + \sqrt{6}}\)
\(\text{} = {2 (4 - \sqrt{6}) \over 16 - 6}\)
\(\text{} = \frac{1}{5} (4 - \sqrt{6})\)
\(\text{} = \frac{4}{5} - \frac{1}{5} \sqrt{6}\)

\({\sqrt{5} \over \sqrt{3} - \sqrt{6}} = {\sqrt{5} \over \sqrt{3} - \sqrt{6}} ⋅ {\sqrt{3} + \sqrt{6} \over \sqrt{3} + \sqrt{6}}\)
\(\text{} = {\sqrt{5} (\sqrt{3} + \sqrt{6}) \over 3 - 6}\)
\(\text{} = -\frac{1}{3} \sqrt{5} (\sqrt{3} + \sqrt{6})\)
\(\text{} = -\frac{1}{3} \sqrt{15} - \frac{1}{3} \sqrt{30}\)