\(\sqrt{12} = \sqrt{4} ⋅ \sqrt{3} = 2 \sqrt{3} \text{.}\)

\(7 \sqrt{75} = 7 ⋅ \sqrt{25} ⋅ \sqrt{3} = 7 ⋅ 5 ⋅ \sqrt{3} = 35 \sqrt{3} \text{.}\)

\(\sqrt{\frac{36}{49}} = {\sqrt{36} \over \sqrt{49}} = \frac{6}{7} \text{.}\)

\({14 \sqrt{56} \over 7 \sqrt{2}} = {14 \over 7} ⋅ {\sqrt{56} \over \sqrt{2}} = 2 \sqrt{28} = 2 ⋅ \sqrt{4} ⋅ \sqrt{7} = 2 ⋅ 2 ⋅ \sqrt{7} = 4 \sqrt{7}\)

\(\sqrt{11\frac{1}{4}} = \sqrt{\frac{45}{4}} = {\sqrt{45} \over \sqrt{4}} = {\sqrt{45} \over 2} = \frac{1}{2} \sqrt{45} = \frac{1}{2} ⋅ 3 ⋅ \sqrt{5} = 1\frac{1}{2} \sqrt{5} \text{.}\)

\(\sqrt{50} + \sqrt{32} = \sqrt{25} ⋅ \sqrt{2} + \sqrt{16} ⋅ \sqrt{2} = 5 \sqrt{2} + 4 \sqrt{2} \text{.}\)

\(5 \sqrt{2} + 4 \sqrt{2} = 9 \sqrt{2} \text{.}\)

\(6 \sqrt{75} + 5 \sqrt{48} = 6 ⋅ \sqrt{25} ⋅ \sqrt{3} + 5 ⋅ \sqrt{16} ⋅ \sqrt{3} \text{.}\)

\(6 ⋅ 5 ⋅ \sqrt{3} + 5 ⋅ 4 ⋅ \sqrt{3} = 30 \sqrt{3} + 20 \sqrt{3} = 50 \sqrt{3} \text{.}\)

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

\(\sqrt{1\frac{19}{45}} = \sqrt{\frac{64}{45}} = {\sqrt{64} \over \sqrt{45}} = {8 \over \sqrt{45}} ⋅ {\sqrt{45} \over \sqrt{45}} = {8 \sqrt{45} \over 45} = \frac{8}{45} \sqrt{45} = \frac{8}{45} ⋅ 3 ⋅ \sqrt{5} = \frac{8}{15} \sqrt{5} \text{.}\)

\(\sqrt{2\frac{1}{2}} = \sqrt{\frac{5}{2}} = {\sqrt{5} \over \sqrt{2}} ⋅ {\sqrt{2} \over \sqrt{2}} = {\sqrt{10} \over 2} = \frac{1}{2} \sqrt{10} \text{.}\)

\(4 \sqrt{10} ⋅ 3 \sqrt{6} = 12 \sqrt{60} = 12 ⋅ \sqrt{4} ⋅ \sqrt{15} = 12 ⋅ 2 ⋅ \sqrt{15} = 24 \sqrt{15}\)