\(\sqrt{175} = \sqrt{25} ⋅ \sqrt{7} = 5 \sqrt{7} \text{.}\)

\(2 \sqrt{500} = 2 ⋅ \sqrt{100} ⋅ \sqrt{5} = 2 ⋅ 10 ⋅ \sqrt{5} = 20 \sqrt{5} \text{.}\)

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

\({16 \sqrt{48} \over 8 \sqrt{6}} = {16 \over 8} ⋅ {\sqrt{48} \over \sqrt{6}} = 2 \sqrt{8} = 2 ⋅ \sqrt{4} ⋅ \sqrt{2} = 2 ⋅ 2 ⋅ \sqrt{2} = 4 \sqrt{2}\)

\(\sqrt{\frac{8}{49}} = {\sqrt{8} \over \sqrt{49}} = {\sqrt{8} \over 7} = \frac{1}{7} \sqrt{8} = \frac{1}{7} ⋅ 2 ⋅ \sqrt{2} = \frac{2}{7} \sqrt{2} \text{.}\)

\(\sqrt{200} + \sqrt{18} = \sqrt{100} ⋅ \sqrt{2} + \sqrt{9} ⋅ \sqrt{2} = 10 \sqrt{2} + 3 \sqrt{2} \text{.}\)

\(10 \sqrt{2} + 3 \sqrt{2} = 13 \sqrt{2} \text{.}\)

\(2 \sqrt{32} - 6 \sqrt{50} = 2 ⋅ \sqrt{16} ⋅ \sqrt{2} - 6 ⋅ \sqrt{25} ⋅ \sqrt{2} \text{.}\)

\(2 ⋅ 4 ⋅ \sqrt{2} - 6 ⋅ 5 ⋅ \sqrt{2} = 8 \sqrt{2} - 30 \sqrt{2} = -22 \sqrt{2} \text{.}\)

\({4 \over 7 \sqrt{3}} = {4 \over 7 \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {4 \sqrt{3} \over 7 ⋅ 3} = \frac{4}{21} \sqrt{3} \text{.}\)

\(\sqrt{2\frac{11}{35}} = \sqrt{\frac{81}{35}} = {\sqrt{81} \over \sqrt{35}} = {9 \over \sqrt{35}} ⋅ {\sqrt{35} \over \sqrt{35}} = {9 \sqrt{35} \over 35} = \frac{9}{35} \sqrt{35} \text{.}\)

\(\sqrt{\frac{2}{3}} = {\sqrt{2} \over \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {\sqrt{6} \over 3} = \frac{1}{3} \sqrt{6} \text{.}\)

\(3 \sqrt{14} ⋅ 2 \sqrt{6} = 6 \sqrt{84} = 6 ⋅ \sqrt{4} ⋅ \sqrt{21} = 6 ⋅ 2 ⋅ \sqrt{21} = 12 \sqrt{21}\)