\(\sqrt{48} + \sqrt{75} = \sqrt{16} ⋅ \sqrt{3} + \sqrt{25} ⋅ \sqrt{3} = 4 \sqrt{3} + 5 \sqrt{3} \text{.}\)

\(4 \sqrt{3} + 5 \sqrt{3} = 9 \sqrt{3} \text{.}\)

\(\sqrt{75} = \sqrt{25} ⋅ \sqrt{3} = 5 \sqrt{3} \text{.}\)

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

\(6 \sqrt{50} - 4 \sqrt{8} = 6 ⋅ \sqrt{25} ⋅ \sqrt{2} - 4 ⋅ \sqrt{4} ⋅ \sqrt{2} \text{.}\)

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

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

\({5 \over 3 \sqrt{2}} = {5 \over 3 \sqrt{2}} ⋅ {\sqrt{2} \over \sqrt{2}} = {5 \sqrt{2} \over 3 ⋅ 2} = \frac{5}{6} \sqrt{2} \text{.}\)

\(\sqrt{15\frac{3}{4}} = \sqrt{\frac{63}{4}} = {\sqrt{63} \over \sqrt{4}} = {\sqrt{63} \over 2} = \frac{1}{2} \sqrt{63} = \frac{1}{2} ⋅ 3 ⋅ \sqrt{7} = 1\frac{1}{2} \sqrt{7} \text{.}\)

\(\sqrt{\frac{9}{62}} = {\sqrt{9} \over \sqrt{62}} = {3 \over \sqrt{62}} ⋅ {\sqrt{62} \over \sqrt{62}} = {3 \sqrt{62} \over 62} = \frac{3}{62} \sqrt{62} \text{.}\)

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

\({15 \sqrt{64} \over 5 \sqrt{8}} = {15 \over 5} ⋅ {\sqrt{64} \over \sqrt{8}} = 3 \sqrt{8} = 3 ⋅ \sqrt{4} ⋅ \sqrt{2} = 3 ⋅ 2 ⋅ \sqrt{2} = 6 \sqrt{2}\)

\(3 \sqrt{5} ⋅ 4 \sqrt{10} = 12 \sqrt{50} = 12 ⋅ \sqrt{25} ⋅ \sqrt{2} = 12 ⋅ 5 ⋅ \sqrt{2} = 60 \sqrt{2}\)