\(\sqrt{32} + \sqrt{200} = \sqrt{16} ⋅ \sqrt{2} + \sqrt{100} ⋅ \sqrt{2} = 4 \sqrt{2} + 10 \sqrt{2} \text{.}\)

\(4 \sqrt{2} + 10 \sqrt{2} = 14 \sqrt{2} \text{.}\)

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

\(4 \sqrt{200} = 4 ⋅ \sqrt{100} ⋅ \sqrt{2} = 4 ⋅ 10 ⋅ \sqrt{2} = 40 \sqrt{2} \text{.}\)

\(2 \sqrt{18} - 5 \sqrt{32} = 2 ⋅ \sqrt{9} ⋅ \sqrt{2} - 5 ⋅ \sqrt{16} ⋅ \sqrt{2} \text{.}\)

\(2 ⋅ 3 ⋅ \sqrt{2} - 5 ⋅ 4 ⋅ \sqrt{2} = 6 \sqrt{2} - 20 \sqrt{2} = -14 \sqrt{2} \text{.}\)

\(\sqrt{\frac{9}{100}} = {\sqrt{9} \over \sqrt{100}} = \frac{3}{10} \text{.}\)

\({9 \over 8 \sqrt{2}} = {9 \over 8 \sqrt{2}} ⋅ {\sqrt{2} \over \sqrt{2}} = {9 \sqrt{2} \over 8 ⋅ 2} = \frac{9}{16} \sqrt{2} \text{.}\)

\(\sqrt{1\frac{6}{25}} = \sqrt{\frac{31}{25}} = {\sqrt{31} \over \sqrt{25}} = {\sqrt{31} \over 5} = \frac{1}{5} \sqrt{31} \text{.}\)

\(\sqrt{\frac{81}{88}} = {\sqrt{81} \over \sqrt{88}} = {9 \over \sqrt{88}} ⋅ {\sqrt{88} \over \sqrt{88}} = {9 \sqrt{88} \over 88} = \frac{9}{88} \sqrt{88} = \frac{9}{88} ⋅ 2 ⋅ \sqrt{22} = \frac{9}{44} \sqrt{22} \text{.}\)

\(\sqrt{22\frac{1}{2}} = \sqrt{\frac{45}{2}} = {\sqrt{45} \over \sqrt{2}} ⋅ {\sqrt{2} \over \sqrt{2}} = {\sqrt{90} \over 2} = \frac{1}{2} \sqrt{90} = \frac{1}{2} ⋅ 3 ⋅ \sqrt{10} = 1\frac{1}{2} \sqrt{10} \text{.}\)

\({16 \sqrt{240} \over 2 \sqrt{6}} = {16 \over 2} ⋅ {\sqrt{240} \over \sqrt{6}} = 8 \sqrt{40} = 8 ⋅ \sqrt{4} ⋅ \sqrt{10} = 8 ⋅ 2 ⋅ \sqrt{10} = 16 \sqrt{10}\)

\(4 \sqrt{10} ⋅ 5 \sqrt{5} = 20 \sqrt{50} = 20 ⋅ \sqrt{25} ⋅ \sqrt{2} = 20 ⋅ 5 ⋅ \sqrt{2} = 100 \sqrt{2}\)