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

\(4 \sqrt{27} = 4 ⋅ \sqrt{9} ⋅ \sqrt{3} = 4 ⋅ 3 ⋅ \sqrt{3} = 12 \sqrt{3} \text{.}\)

\(\sqrt{1\frac{32}{49}} = \sqrt{\frac{81}{49}} = {\sqrt{81} \over \sqrt{49}} = \frac{9}{7} = 1\frac{2}{7} \text{.}\)

\({2 \sqrt{56} \over \sqrt{2}} = 2 ⋅ {\sqrt{56} \over \sqrt{2}} = 2 \sqrt{28} = 2 ⋅ \sqrt{4} ⋅ \sqrt{7} = 2 ⋅ 2 ⋅ \sqrt{7} = 4 \sqrt{7}\)

\(\sqrt{1\frac{31}{49}} = \sqrt{\frac{80}{49}} = {\sqrt{80} \over \sqrt{49}} = {\sqrt{80} \over 7} = \frac{1}{7} \sqrt{80} = \frac{1}{7} ⋅ 4 ⋅ \sqrt{5} = \frac{4}{7} \sqrt{5} \text{.}\)

\(\sqrt{80} + \sqrt{125} = \sqrt{16} ⋅ \sqrt{5} + \sqrt{25} ⋅ \sqrt{5} = 4 \sqrt{5} + 5 \sqrt{5} \text{.}\)

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

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

\(6 ⋅ 4 ⋅ \sqrt{2} + 2 ⋅ 5 ⋅ \sqrt{2} = 24 \sqrt{2} + 10 \sqrt{2} = 34 \sqrt{2} \text{.}\)

\({9 \over 5 \sqrt{3}} = {9 \over 5 \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {9 \sqrt{3} \over 5 ⋅ 3} = \frac{3}{5} \sqrt{3} \text{.}\)

\(\sqrt{\frac{9}{80}} = {\sqrt{9} \over \sqrt{80}} = {3 \over \sqrt{80}} ⋅ {\sqrt{80} \over \sqrt{80}} = {3 \sqrt{80} \over 80} = \frac{3}{80} \sqrt{80} = \frac{3}{80} ⋅ 4 ⋅ \sqrt{5} = \frac{3}{20} \sqrt{5} \text{.}\)

\(\sqrt{1\frac{2}{5}} = \sqrt{\frac{7}{5}} = {\sqrt{7} \over \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {\sqrt{35} \over 5} = \frac{1}{5} \sqrt{35} \text{.}\)