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

\(-2 \sqrt{27} = -2 ⋅ \sqrt{9} ⋅ \sqrt{3} = -2 ⋅ 3 ⋅ \sqrt{3} = -6 \sqrt{3} \text{.}\)

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

\({3 \sqrt{84} \over \sqrt{7}} = 3 ⋅ {\sqrt{84} \over \sqrt{7}} = 3 \sqrt{12} = 3 ⋅ \sqrt{4} ⋅ \sqrt{3} = 3 ⋅ 2 ⋅ \sqrt{3} = 6 \sqrt{3}\)

\(\sqrt{\frac{27}{64}} = {\sqrt{27} \over \sqrt{64}} = {\sqrt{27} \over 8} = \frac{1}{8} \sqrt{27} = \frac{1}{8} ⋅ 3 ⋅ \sqrt{3} = \frac{3}{8} \sqrt{3} \text{.}\)

\(\sqrt{50} + \sqrt{18} = \sqrt{25} ⋅ \sqrt{2} + \sqrt{9} ⋅ \sqrt{2} = 5 \sqrt{2} + 3 \sqrt{2} \text{.}\)

\(5 \sqrt{2} + 3 \sqrt{2} = 8 \sqrt{2} \text{.}\)

\(6 \sqrt{12} - 3 \sqrt{300} = 6 ⋅ \sqrt{4} ⋅ \sqrt{3} - 3 ⋅ \sqrt{100} ⋅ \sqrt{3} \text{.}\)

\(6 ⋅ 2 ⋅ \sqrt{3} - 3 ⋅ 10 ⋅ \sqrt{3} = 12 \sqrt{3} - 30 \sqrt{3} = -18 \sqrt{3} \text{.}\)

\({8 \over 3 \sqrt{2}} = {8 \over 3 \sqrt{2}} ⋅ {\sqrt{2} \over \sqrt{2}} = {8 \sqrt{2} \over 3 ⋅ 2} = 1\frac{1}{3} \sqrt{2} \text{.}\)

\(\sqrt{8\frac{1}{6}} = \sqrt{\frac{49}{6}} = {\sqrt{49} \over \sqrt{6}} = {7 \over \sqrt{6}} ⋅ {\sqrt{6} \over \sqrt{6}} = {7 \sqrt{6} \over 6} = 1\frac{1}{6} \sqrt{6} \text{.}\)

\(\sqrt{1\frac{1}{5}} = \sqrt{\frac{6}{5}} = {\sqrt{6} \over \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {\sqrt{30} \over 5} = \frac{1}{5} \sqrt{30} \text{.}\)

\(5 \sqrt{6} ⋅ 3 \sqrt{14} = 15 \sqrt{84} = 15 ⋅ \sqrt{4} ⋅ \sqrt{21} = 15 ⋅ 2 ⋅ \sqrt{21} = 30 \sqrt{21}\)