\(\sqrt{18} = \sqrt{9} ⋅ \sqrt{2} = 3 \sqrt{2} \text{.}\)

\(2 \sqrt{63} = 2 ⋅ \sqrt{9} ⋅ \sqrt{7} = 2 ⋅ 3 ⋅ \sqrt{7} = 6 \sqrt{7} \text{.}\)

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

\({54 \sqrt{192} \over 9 \sqrt{8}} = {54 \over 9} ⋅ {\sqrt{192} \over \sqrt{8}} = 6 \sqrt{24} = 6 ⋅ \sqrt{4} ⋅ \sqrt{6} = 6 ⋅ 2 ⋅ \sqrt{6} = 12 \sqrt{6}\)

\(\sqrt{\frac{13}{64}} = {\sqrt{13} \over \sqrt{64}} = {\sqrt{13} \over 8} = \frac{1}{8} \sqrt{13} \text{.}\)

\(\sqrt{80} + \sqrt{500} = \sqrt{16} ⋅ \sqrt{5} + \sqrt{100} ⋅ \sqrt{5} = 4 \sqrt{5} + 10 \sqrt{5} \text{.}\)

\(4 \sqrt{5} + 10 \sqrt{5} = 14 \sqrt{5} \text{.}\)

\(7 \sqrt{27} - 3 \sqrt{75} = 7 ⋅ \sqrt{9} ⋅ \sqrt{3} - 3 ⋅ \sqrt{25} ⋅ \sqrt{3} \text{.}\)

\(7 ⋅ 3 ⋅ \sqrt{3} - 3 ⋅ 5 ⋅ \sqrt{3} = 21 \sqrt{3} - 15 \sqrt{3} = 6 \sqrt{3} \text{.}\)

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

\(\sqrt{\frac{9}{13}} = {\sqrt{9} \over \sqrt{13}} = {3 \over \sqrt{13}} ⋅ {\sqrt{13} \over \sqrt{13}} = {3 \sqrt{13} \over 13} = \frac{3}{13} \sqrt{13} \text{.}\)

\(\sqrt{\frac{5}{14}} = {\sqrt{5} \over \sqrt{14}} ⋅ {\sqrt{14} \over \sqrt{14}} = {\sqrt{70} \over 14} = \frac{1}{14} \sqrt{70} \text{.}\)