\(\sqrt{63} = \sqrt{9} ⋅ \sqrt{7} = 3 \sqrt{7} \text{.}\)

\(-4 \sqrt{200} = -4 ⋅ \sqrt{100} ⋅ \sqrt{2} = -4 ⋅ 10 ⋅ \sqrt{2} = -40 \sqrt{2} \text{.}\)

\(\sqrt{1\frac{19}{81}} = \sqrt{\frac{100}{81}} = {\sqrt{100} \over \sqrt{81}} = \frac{10}{9} = 1\frac{1}{9} \text{.}\)

\({30 \sqrt{56} \over 5 \sqrt{7}} = {30 \over 5} ⋅ {\sqrt{56} \over \sqrt{7}} = 6 \sqrt{8} = 6 ⋅ \sqrt{4} ⋅ \sqrt{2} = 6 ⋅ 2 ⋅ \sqrt{2} = 12 \sqrt{2}\)

\(\sqrt{5\frac{7}{9}} = \sqrt{\frac{52}{9}} = {\sqrt{52} \over \sqrt{9}} = {\sqrt{52} \over 3} = \frac{1}{3} \sqrt{52} = \frac{1}{3} ⋅ 2 ⋅ \sqrt{13} = \frac{2}{3} \sqrt{13} \text{.}\)

\(\sqrt{27} + \sqrt{300} = \sqrt{9} ⋅ \sqrt{3} + \sqrt{100} ⋅ \sqrt{3} = 3 \sqrt{3} + 10 \sqrt{3} \text{.}\)

\(3 \sqrt{3} + 10 \sqrt{3} = 13 \sqrt{3} \text{.}\)

\(2 \sqrt{27} + 3 \sqrt{300} = 2 ⋅ \sqrt{9} ⋅ \sqrt{3} + 3 ⋅ \sqrt{100} ⋅ \sqrt{3} \text{.}\)

\(2 ⋅ 3 ⋅ \sqrt{3} + 3 ⋅ 10 ⋅ \sqrt{3} = 6 \sqrt{3} + 30 \sqrt{3} = 36 \sqrt{3} \text{.}\)

\({2 \over 7 \sqrt{3}} = {2 \over 7 \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {2 \sqrt{3} \over 7 ⋅ 3} = \frac{2}{21} \sqrt{3} \text{.}\)

\(\sqrt{2\frac{1}{40}} = \sqrt{\frac{81}{40}} = {\sqrt{81} \over \sqrt{40}} = {9 \over \sqrt{40}} ⋅ {\sqrt{40} \over \sqrt{40}} = {9 \sqrt{40} \over 40} = \frac{9}{40} \sqrt{40} = \frac{9}{40} ⋅ 2 ⋅ \sqrt{10} = \frac{9}{20} \sqrt{10} \text{.}\)

\(\sqrt{9\frac{3}{8}} = \sqrt{\frac{75}{8}} = {\sqrt{75} \over \sqrt{8}} ⋅ {\sqrt{8} \over \sqrt{8}} = {\sqrt{600} \over 8} = \frac{1}{8} \sqrt{600} = \frac{1}{8} ⋅ 10 ⋅ \sqrt{6} = 1\frac{1}{4} \sqrt{6} \text{.}\)

\(3 \sqrt{15} ⋅ 5 \sqrt{6} = 15 \sqrt{90} = 15 ⋅ \sqrt{9} ⋅ \sqrt{10} = 15 ⋅ 3 ⋅ \sqrt{10} = 45 \sqrt{10}\)