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

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

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

\(5 \sqrt{300} = 5 ⋅ \sqrt{100} ⋅ \sqrt{3} = 5 ⋅ 10 ⋅ \sqrt{3} = 50 \sqrt{3} \text{.}\)

\(5 \sqrt{27} - 3 \sqrt{75} = 5 ⋅ \sqrt{9} ⋅ \sqrt{3} - 3 ⋅ \sqrt{25} ⋅ \sqrt{3} \text{.}\)

\(5 ⋅ 3 ⋅ \sqrt{3} - 3 ⋅ 5 ⋅ \sqrt{3} = 15 \sqrt{3} - 15 \sqrt{3} = 0 \sqrt{3} \text{.}\)

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

\({8 \over 3 \sqrt{5}} = {8 \over 3 \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {8 \sqrt{5} \over 3 ⋅ 5} = \frac{8}{15} \sqrt{5} \text{.}\)

\(\sqrt{\frac{44}{49}} = {\sqrt{44} \over \sqrt{49}} = {\sqrt{44} \over 7} = \frac{1}{7} \sqrt{44} = \frac{1}{7} ⋅ 2 ⋅ \sqrt{11} = \frac{2}{7} \sqrt{11} \text{.}\)

\(\sqrt{2\frac{6}{47}} = \sqrt{\frac{100}{47}} = {\sqrt{100} \over \sqrt{47}} = {10 \over \sqrt{47}} ⋅ {\sqrt{47} \over \sqrt{47}} = {10 \sqrt{47} \over 47} = \frac{10}{47} \sqrt{47} \text{.}\)

\(\sqrt{\frac{2}{19}} = {\sqrt{2} \over \sqrt{19}} ⋅ {\sqrt{19} \over \sqrt{19}} = {\sqrt{38} \over 19} = \frac{1}{19} \sqrt{38} \text{.}\)

\({15 \sqrt{84} \over 3 \sqrt{3}} = {15 \over 3} ⋅ {\sqrt{84} \over \sqrt{3}} = 5 \sqrt{28} = 5 ⋅ \sqrt{4} ⋅ \sqrt{7} = 5 ⋅ 2 ⋅ \sqrt{7} = 10 \sqrt{7}\)

\(4 \sqrt{6} ⋅ 5 \sqrt{2} = 20 \sqrt{12} = 20 ⋅ \sqrt{4} ⋅ \sqrt{3} = 20 ⋅ 2 ⋅ \sqrt{3} = 40 \sqrt{3}\)