\(\sqrt{80} + \sqrt{20} = \sqrt{16} ⋅ \sqrt{5} + \sqrt{4} ⋅ \sqrt{5} = 4 \sqrt{5} + 2 \sqrt{5} \text{.}\)

\(4 \sqrt{5} + 2 \sqrt{5} = 6 \sqrt{5} \text{.}\)

\(\sqrt{27} = \sqrt{9} ⋅ \sqrt{3} = 3 \sqrt{3} \text{.}\)

\(4 \sqrt{12} = 4 ⋅ \sqrt{4} ⋅ \sqrt{3} = 4 ⋅ 2 ⋅ \sqrt{3} = 8 \sqrt{3} \text{.}\)

\(6 \sqrt{27} + 2 \sqrt{75} = 6 ⋅ \sqrt{9} ⋅ \sqrt{3} + 2 ⋅ \sqrt{25} ⋅ \sqrt{3} \text{.}\)

\(6 ⋅ 3 ⋅ \sqrt{3} + 2 ⋅ 5 ⋅ \sqrt{3} = 18 \sqrt{3} + 10 \sqrt{3} = 28 \sqrt{3} \text{.}\)

\(\sqrt{\frac{1}{25}} = {\sqrt{1} \over \sqrt{25}} = \frac{1}{5} \text{.}\)

\({2 \over 9 \sqrt{7}} = {2 \over 9 \sqrt{7}} ⋅ {\sqrt{7} \over \sqrt{7}} = {2 \sqrt{7} \over 9 ⋅ 7} = \frac{2}{63} \sqrt{7} \text{.}\)

\(\sqrt{1\frac{19}{25}} = \sqrt{\frac{44}{25}} = {\sqrt{44} \over \sqrt{25}} = {\sqrt{44} \over 5} = \frac{1}{5} \sqrt{44} = \frac{1}{5} ⋅ 2 ⋅ \sqrt{11} = \frac{2}{5} \sqrt{11} \text{.}\)

\(\sqrt{1\frac{8}{73}} = \sqrt{\frac{81}{73}} = {\sqrt{81} \over \sqrt{73}} = {9 \over \sqrt{73}} ⋅ {\sqrt{73} \over \sqrt{73}} = {9 \sqrt{73} \over 73} = \frac{9}{73} \sqrt{73} \text{.}\)

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

\({18 \sqrt{72} \over 9 \sqrt{6}} = {18 \over 9} ⋅ {\sqrt{72} \over \sqrt{6}} = 2 \sqrt{12} = 2 ⋅ \sqrt{4} ⋅ \sqrt{3} = 2 ⋅ 2 ⋅ \sqrt{3} = 4 \sqrt{3}\)

\(4 \sqrt{6} ⋅ 5 \sqrt{14} = 20 \sqrt{84} = 20 ⋅ \sqrt{4} ⋅ \sqrt{21} = 20 ⋅ 2 ⋅ \sqrt{21} = 40 \sqrt{21}\)

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

\({4 \sqrt{5} \over \sqrt{2} - \sqrt{3}} = {4 \sqrt{5} \over \sqrt{2} - \sqrt{3}} ⋅ {\sqrt{2} + \sqrt{3} \over \sqrt{2} + \sqrt{3}}\)
\(\text{} = {4 \sqrt{5} (\sqrt{2} + \sqrt{3}) \over 2 - 3}\)
\(\text{} = -4 \sqrt{5} (\sqrt{2} + \sqrt{3})\)
\(\text{} = -4 \sqrt{10} - 4 \sqrt{15}\)