\(\sqrt{500} + \sqrt{45} = \sqrt{100} ⋅ \sqrt{5} + \sqrt{9} ⋅ \sqrt{5} = 10 \sqrt{5} + 3 \sqrt{5} \text{.}\)

\(10 \sqrt{5} + 3 \sqrt{5} = 13 \sqrt{5} \text{.}\)

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

\(7 \sqrt{50} = 7 ⋅ \sqrt{25} ⋅ \sqrt{2} = 7 ⋅ 5 ⋅ \sqrt{2} = 35 \sqrt{2} \text{.}\)

\(7 \sqrt{500} - 2 \sqrt{125} = 7 ⋅ \sqrt{100} ⋅ \sqrt{5} - 2 ⋅ \sqrt{25} ⋅ \sqrt{5} \text{.}\)

\(7 ⋅ 10 ⋅ \sqrt{5} - 2 ⋅ 5 ⋅ \sqrt{5} = 70 \sqrt{5} - 10 \sqrt{5} = 60 \sqrt{5} \text{.}\)

\(\sqrt{\frac{49}{100}} = {\sqrt{49} \over \sqrt{100}} = \frac{7}{10} \text{.}\)

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

\(\sqrt{1\frac{5}{49}} = \sqrt{\frac{54}{49}} = {\sqrt{54} \over \sqrt{49}} = {\sqrt{54} \over 7} = \frac{1}{7} \sqrt{54} = \frac{1}{7} ⋅ 3 ⋅ \sqrt{6} = \frac{3}{7} \sqrt{6} \text{.}\)

\(\sqrt{\frac{16}{55}} = {\sqrt{16} \over \sqrt{55}} = {4 \over \sqrt{55}} ⋅ {\sqrt{55} \over \sqrt{55}} = {4 \sqrt{55} \over 55} = \frac{4}{55} \sqrt{55} \text{.}\)

\(\sqrt{\frac{2}{37}} = {\sqrt{2} \over \sqrt{37}} ⋅ {\sqrt{37} \over \sqrt{37}} = {\sqrt{74} \over 37} = \frac{1}{37} \sqrt{74} \text{.}\)

\({15 \sqrt{144} \over 3 \sqrt{6}} = {15 \over 3} ⋅ {\sqrt{144} \over \sqrt{6}} = 5 \sqrt{24} = 5 ⋅ \sqrt{4} ⋅ \sqrt{6} = 5 ⋅ 2 ⋅ \sqrt{6} = 10 \sqrt{6}\)

\(5 \sqrt{7} ⋅ 4 \sqrt{14} = 20 \sqrt{98} = 20 ⋅ \sqrt{49} ⋅ \sqrt{2} = 20 ⋅ 7 ⋅ \sqrt{2} = 140 \sqrt{2}\)

\({5 \over 4 + \sqrt{2}} = {5 \over 4 + \sqrt{2}} ⋅ {4 - \sqrt{2} \over 4 - \sqrt{2}}\)
\(\text{} = {5 (4 + \sqrt{2}) \over 16 - 2}\)
\(\text{} = \frac{5}{14} (4 + \sqrt{2})\)
\(\text{} = 1\frac{3}{7} + \frac{5}{14} \sqrt{2}\)

\({6 \sqrt{5} \over \sqrt{2} - \sqrt{3}} = {6 \sqrt{5} \over \sqrt{2} - \sqrt{3}} ⋅ {\sqrt{2} + \sqrt{3} \over \sqrt{2} + \sqrt{3}}\)
\(\text{} = {6 \sqrt{5} (\sqrt{2} + \sqrt{3}) \over 2 - 3}\)
\(\text{} = -6 \sqrt{5} (\sqrt{2} + \sqrt{3})\)
\(\text{} = -6 \sqrt{10} - 6 \sqrt{15}\)