\(\sqrt{45} + \sqrt{500} = \sqrt{9} ⋅ \sqrt{5} + \sqrt{100} ⋅ \sqrt{5} = 3 \sqrt{5} + 10 \sqrt{5} \text{.}\)

\(3 \sqrt{5} + 10 \sqrt{5} = 13 \sqrt{5} \text{.}\)

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

\(6 \sqrt{500} = 6 ⋅ \sqrt{100} ⋅ \sqrt{5} = 6 ⋅ 10 ⋅ \sqrt{5} = 60 \sqrt{5} \text{.}\)

\(2 \sqrt{18} - 3 \sqrt{8} = 2 ⋅ \sqrt{9} ⋅ \sqrt{2} - 3 ⋅ \sqrt{4} ⋅ \sqrt{2} \text{.}\)

\(2 ⋅ 3 ⋅ \sqrt{2} - 3 ⋅ 2 ⋅ \sqrt{2} = 6 \sqrt{2} - 6 \sqrt{2} = 0 \sqrt{2} \text{.}\)

\(\sqrt{\frac{4}{25}} = {\sqrt{4} \over \sqrt{25}} = \frac{2}{5} \text{.}\)

\({3 \over 2 \sqrt{5}} = {3 \over 2 \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {3 \sqrt{5} \over 2 ⋅ 5} = \frac{3}{10} \sqrt{5} \text{.}\)

\(\sqrt{\frac{55}{81}} = {\sqrt{55} \over \sqrt{81}} = {\sqrt{55} \over 9} = \frac{1}{9} \sqrt{55} \text{.}\)

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

\(\sqrt{3\frac{3}{5}} = \sqrt{\frac{18}{5}} = {\sqrt{18} \over \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {\sqrt{90} \over 5} = \frac{1}{5} \sqrt{90} = \frac{1}{5} ⋅ 3 ⋅ \sqrt{10} = \frac{3}{5} \sqrt{10} \text{.}\)

\({16 \sqrt{280} \over 8 \sqrt{10}} = {16 \over 8} ⋅ {\sqrt{280} \over \sqrt{10}} = 2 \sqrt{28} = 2 ⋅ \sqrt{4} ⋅ \sqrt{7} = 2 ⋅ 2 ⋅ \sqrt{7} = 4 \sqrt{7}\)

\(5 \sqrt{2} ⋅ 3 \sqrt{10} = 15 \sqrt{20} = 15 ⋅ \sqrt{4} ⋅ \sqrt{5} = 15 ⋅ 2 ⋅ \sqrt{5} = 30 \sqrt{5}\)

\({-4 \over 1 - \sqrt{3}} = {-4 \over 1 - \sqrt{3}} ⋅ {1 + \sqrt{3} \over 1 + \sqrt{3}}\)
\(\text{} = {-4 (1 - \sqrt{3}) \over 1 - 3}\)
\(\text{} = 2 (1 - \sqrt{3})\)
\(\text{} = 2 - 2 \sqrt{3}\)

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