\(\sqrt{50} = \sqrt{25} ⋅ \sqrt{2} = 5 \sqrt{2} \text{.}\)

\(-3 \sqrt{500} = -3 ⋅ \sqrt{100} ⋅ \sqrt{5} = -3 ⋅ 10 ⋅ \sqrt{5} = -30 \sqrt{5} \text{.}\)

\(\sqrt{7\frac{1}{9}} = \sqrt{\frac{64}{9}} = {\sqrt{64} \over \sqrt{9}} = \frac{8}{3} = 2\frac{2}{3} \text{.}\)

\({8 \sqrt{140} \over 2 \sqrt{7}} = {8 \over 2} ⋅ {\sqrt{140} \over \sqrt{7}} = 4 \sqrt{20} = 4 ⋅ \sqrt{4} ⋅ \sqrt{5} = 4 ⋅ 2 ⋅ \sqrt{5} = 8 \sqrt{5}\)

\(\sqrt{1\frac{36}{49}} = \sqrt{\frac{85}{49}} = {\sqrt{85} \over \sqrt{49}} = {\sqrt{85} \over 7} = \frac{1}{7} \sqrt{85} \text{.}\)

\(\sqrt{125} + \sqrt{500} = \sqrt{25} ⋅ \sqrt{5} + \sqrt{100} ⋅ \sqrt{5} = 5 \sqrt{5} + 10 \sqrt{5} \text{.}\)

\(5 \sqrt{5} + 10 \sqrt{5} = 15 \sqrt{5} \text{.}\)

\(4 \sqrt{50} + 6 \sqrt{32} = 4 ⋅ \sqrt{25} ⋅ \sqrt{2} + 6 ⋅ \sqrt{16} ⋅ \sqrt{2} \text{.}\)

\(4 ⋅ 5 ⋅ \sqrt{2} + 6 ⋅ 4 ⋅ \sqrt{2} = 20 \sqrt{2} + 24 \sqrt{2} = 44 \sqrt{2} \text{.}\)

\({5 \over 4 \sqrt{3}} = {5 \over 4 \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {5 \sqrt{3} \over 4 ⋅ 3} = \frac{5}{12} \sqrt{3} \text{.}\)

\(\sqrt{2\frac{1}{12}} = \sqrt{\frac{25}{12}} = {\sqrt{25} \over \sqrt{12}} = {5 \over \sqrt{12}} ⋅ {\sqrt{12} \over \sqrt{12}} = {5 \sqrt{12} \over 12} = \frac{5}{12} \sqrt{12} = \frac{5}{12} ⋅ 2 ⋅ \sqrt{3} = \frac{5}{6} \sqrt{3} \text{.}\)

\(\sqrt{3\frac{1}{3}} = \sqrt{\frac{10}{3}} = {\sqrt{10} \over \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {\sqrt{30} \over 3} = \frac{1}{3} \sqrt{30} \text{.}\)