\(\sqrt{125} + \sqrt{20} = \sqrt{25} ⋅ \sqrt{5} + \sqrt{4} ⋅ \sqrt{5} = 5 \sqrt{5} + 2 \sqrt{5} \text{.}\)

\(5 \sqrt{5} + 2 \sqrt{5} = 7 \sqrt{5} \text{.}\)

\(\sqrt{28} = \sqrt{4} ⋅ \sqrt{7} = 2 \sqrt{7} \text{.}\)

\(-3 \sqrt{80} = -3 ⋅ \sqrt{16} ⋅ \sqrt{5} = -3 ⋅ 4 ⋅ \sqrt{5} = -12 \sqrt{5} \text{.}\)

\(4 \sqrt{12} - 2 \sqrt{75} = 4 ⋅ \sqrt{4} ⋅ \sqrt{3} - 2 ⋅ \sqrt{25} ⋅ \sqrt{3} \text{.}\)

\(4 ⋅ 2 ⋅ \sqrt{3} - 2 ⋅ 5 ⋅ \sqrt{3} = 8 \sqrt{3} - 10 \sqrt{3} = -2 \sqrt{3} \text{.}\)

\(\sqrt{2\frac{1}{4}} = \sqrt{\frac{9}{4}} = {\sqrt{9} \over \sqrt{4}} = \frac{3}{2} = 1\frac{1}{2} \text{.}\)

\({3 \over 8 \sqrt{7}} = {3 \over 8 \sqrt{7}} ⋅ {\sqrt{7} \over \sqrt{7}} = {3 \sqrt{7} \over 8 ⋅ 7} = \frac{3}{56} \sqrt{7} \text{.}\)

\(\sqrt{8\frac{2}{9}} = \sqrt{\frac{74}{9}} = {\sqrt{74} \over \sqrt{9}} = {\sqrt{74} \over 3} = \frac{1}{3} \sqrt{74} \text{.}\)

\(\sqrt{3\frac{4}{7}} = \sqrt{\frac{25}{7}} = {\sqrt{25} \over \sqrt{7}} = {5 \over \sqrt{7}} ⋅ {\sqrt{7} \over \sqrt{7}} = {5 \sqrt{7} \over 7} = \frac{5}{7} \sqrt{7} \text{.}\)

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

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

\(2 \sqrt{2} ⋅ 3 \sqrt{10} = 6 \sqrt{20} = 6 ⋅ \sqrt{4} ⋅ \sqrt{5} = 6 ⋅ 2 ⋅ \sqrt{5} = 12 \sqrt{5}\)

\({-5 \over 6 + \sqrt{2}} = {-5 \over 6 + \sqrt{2}} ⋅ {6 - \sqrt{2} \over 6 - \sqrt{2}}\)
\(\text{} = {-5 (6 + \sqrt{2}) \over 36 - 2}\)
\(\text{} = -\frac{5}{34} (6 + \sqrt{2})\)
\(\text{} = -\frac{15}{17} - \frac{5}{34} \sqrt{2}\)

\({\sqrt{6} \over \sqrt{3} - \sqrt{2}} = {\sqrt{6} \over \sqrt{3} - \sqrt{2}} ⋅ {\sqrt{3} + \sqrt{2} \over \sqrt{3} + \sqrt{2}}\)
\(\text{} = {\sqrt{6} (\sqrt{3} + \sqrt{2}) \over 3 - 2}\)
\(\text{} = \sqrt{6} (\sqrt{3} + \sqrt{2})\)
\(\text{} = \sqrt{18} + \sqrt{12}\)
\(\text{} = 3 \sqrt{2} + 2 \sqrt{3}\)