\(\sqrt{500} + \sqrt{80} = \sqrt{100} ⋅ \sqrt{5} + \sqrt{16} ⋅ \sqrt{5} = 10 \sqrt{5} + 4 \sqrt{5} \text{.}\)

\(10 \sqrt{5} + 4 \sqrt{5} = 14 \sqrt{5} \text{.}\)

\(\sqrt{18} = \sqrt{9} ⋅ \sqrt{2} = 3 \sqrt{2} \text{.}\)

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

\(7 \sqrt{12} + 4 \sqrt{27} = 7 ⋅ \sqrt{4} ⋅ \sqrt{3} + 4 ⋅ \sqrt{9} ⋅ \sqrt{3} \text{.}\)

\(7 ⋅ 2 ⋅ \sqrt{3} + 4 ⋅ 3 ⋅ \sqrt{3} = 14 \sqrt{3} + 12 \sqrt{3} = 26 \sqrt{3} \text{.}\)

\(\sqrt{\frac{1}{49}} = {\sqrt{1} \over \sqrt{49}} = \frac{1}{7} \text{.}\)

\({6 \over 7 \sqrt{5}} = {6 \over 7 \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {6 \sqrt{5} \over 7 ⋅ 5} = \frac{6}{35} \sqrt{5} \text{.}\)

\(\sqrt{5\frac{3}{4}} = \sqrt{\frac{23}{4}} = {\sqrt{23} \over \sqrt{4}} = {\sqrt{23} \over 2} = \frac{1}{2} \sqrt{23} \text{.}\)

\(\sqrt{\frac{1}{27}} = {\sqrt{1} \over \sqrt{27}} = {1 \over \sqrt{27}} ⋅ {\sqrt{27} \over \sqrt{27}} = {\sqrt{27} \over 27} = \frac{1}{27} \sqrt{27} = \frac{1}{27} ⋅ 3 ⋅ \sqrt{3} = \frac{1}{9} \sqrt{3} \text{.}\)

\(\sqrt{10\frac{1}{3}} = \sqrt{\frac{31}{3}} = {\sqrt{31} \over \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {\sqrt{93} \over 3} = \frac{1}{3} \sqrt{93} \text{.}\)

\({54 \sqrt{72} \over 9 \sqrt{3}} = {54 \over 9} ⋅ {\sqrt{72} \over \sqrt{3}} = 6 \sqrt{24} = 6 ⋅ \sqrt{4} ⋅ \sqrt{6} = 6 ⋅ 2 ⋅ \sqrt{6} = 12 \sqrt{6}\)

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

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

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