\(\sqrt{32} + \sqrt{8} = \sqrt{16} ⋅ \sqrt{2} + \sqrt{4} ⋅ \sqrt{2} = 4 \sqrt{2} + 2 \sqrt{2} \text{.}\)

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

\(\sqrt{63} = \sqrt{9} ⋅ \sqrt{7} = 3 \sqrt{7} \text{.}\)

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

\(3 \sqrt{18} - 5 \sqrt{200} = 3 ⋅ \sqrt{9} ⋅ \sqrt{2} - 5 ⋅ \sqrt{100} ⋅ \sqrt{2} \text{.}\)

\(3 ⋅ 3 ⋅ \sqrt{2} - 5 ⋅ 10 ⋅ \sqrt{2} = 9 \sqrt{2} - 50 \sqrt{2} = -41 \sqrt{2} \text{.}\)

\(\sqrt{5\frac{4}{9}} = \sqrt{\frac{49}{9}} = {\sqrt{49} \over \sqrt{9}} = \frac{7}{3} = 2\frac{1}{3} \text{.}\)

\({7 \over 6 \sqrt{5}} = {7 \over 6 \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {7 \sqrt{5} \over 6 ⋅ 5} = \frac{7}{30} \sqrt{5} \text{.}\)

\(\sqrt{\frac{23}{81}} = {\sqrt{23} \over \sqrt{81}} = {\sqrt{23} \over 9} = \frac{1}{9} \sqrt{23} \text{.}\)

\(\sqrt{1\frac{10}{71}} = \sqrt{\frac{81}{71}} = {\sqrt{81} \over \sqrt{71}} = {9 \over \sqrt{71}} ⋅ {\sqrt{71} \over \sqrt{71}} = {9 \sqrt{71} \over 71} = \frac{9}{71} \sqrt{71} \text{.}\)

\(\sqrt{\frac{2}{75}} = {\sqrt{2} \over \sqrt{75}} ⋅ {\sqrt{75} \over \sqrt{75}} = {\sqrt{150} \over 75} = \frac{1}{75} \sqrt{150} = \frac{1}{75} ⋅ 5 ⋅ \sqrt{6} = \frac{1}{15} \sqrt{6} \text{.}\)

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

\(2 \sqrt{10} ⋅ 3 \sqrt{5} = 6 \sqrt{50} = 6 ⋅ \sqrt{25} ⋅ \sqrt{2} = 6 ⋅ 5 ⋅ \sqrt{2} = 30 \sqrt{2}\)

\({-2 \over 3 - \sqrt{5}} = {-2 \over 3 - \sqrt{5}} ⋅ {3 + \sqrt{5} \over 3 + \sqrt{5}}\)
\(\text{} = {-2 (3 - \sqrt{5}) \over 9 - 5}\)
\(\text{} = -\frac{1}{2} (3 - \sqrt{5})\)
\(\text{} = -1\frac{1}{2} + \frac{1}{2} \sqrt{5}\)

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