\(\sqrt{18} + \sqrt{8} = \sqrt{9} ⋅ \sqrt{2} + \sqrt{4} ⋅ \sqrt{2} = 3 \sqrt{2} + 2 \sqrt{2} \text{.}\)

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

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

\(-4 \sqrt{175} = -4 ⋅ \sqrt{25} ⋅ \sqrt{7} = -4 ⋅ 5 ⋅ \sqrt{7} = -20 \sqrt{7} \text{.}\)

\(3 \sqrt{20} + 7 \sqrt{80} = 3 ⋅ \sqrt{4} ⋅ \sqrt{5} + 7 ⋅ \sqrt{16} ⋅ \sqrt{5} \text{.}\)

\(3 ⋅ 2 ⋅ \sqrt{5} + 7 ⋅ 4 ⋅ \sqrt{5} = 6 \sqrt{5} + 28 \sqrt{5} = 34 \sqrt{5} \text{.}\)

\(\sqrt{5\frac{1}{16}} = \sqrt{\frac{81}{16}} = {\sqrt{81} \over \sqrt{16}} = \frac{9}{4} = 2\frac{1}{4} \text{.}\)

\({3 \over 4 \sqrt{3}} = {3 \over 4 \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {3 \sqrt{3} \over 4 ⋅ 3} = \frac{1}{4} \sqrt{3} \text{.}\)

\(\sqrt{1\frac{16}{25}} = \sqrt{\frac{41}{25}} = {\sqrt{41} \over \sqrt{25}} = {\sqrt{41} \over 5} = \frac{1}{5} \sqrt{41} \text{.}\)

\(\sqrt{\frac{1}{60}} = {\sqrt{1} \over \sqrt{60}} = {1 \over \sqrt{60}} ⋅ {\sqrt{60} \over \sqrt{60}} = {\sqrt{60} \over 60} = \frac{1}{60} \sqrt{60} = \frac{1}{60} ⋅ 2 ⋅ \sqrt{15} = \frac{1}{30} \sqrt{15} \text{.}\)

\(\sqrt{\frac{2}{31}} = {\sqrt{2} \over \sqrt{31}} ⋅ {\sqrt{31} \over \sqrt{31}} = {\sqrt{62} \over 31} = \frac{1}{31} \sqrt{62} \text{.}\)

\({6 \sqrt{168} \over 3 \sqrt{6}} = {6 \over 3} ⋅ {\sqrt{168} \over \sqrt{6}} = 2 \sqrt{28} = 2 ⋅ \sqrt{4} ⋅ \sqrt{7} = 2 ⋅ 2 ⋅ \sqrt{7} = 4 \sqrt{7}\)

\(4 \sqrt{15} ⋅ 5 \sqrt{6} = 20 \sqrt{90} = 20 ⋅ \sqrt{9} ⋅ \sqrt{10} = 20 ⋅ 3 ⋅ \sqrt{10} = 60 \sqrt{10}\)

\({-3 \over 6 - \sqrt{5}} = {-3 \over 6 - \sqrt{5}} ⋅ {6 + \sqrt{5} \over 6 + \sqrt{5}}\)
\(\text{} = {-3 (6 - \sqrt{5}) \over 36 - 5}\)
\(\text{} = -\frac{3}{31} (6 - \sqrt{5})\)
\(\text{} = -\frac{18}{31} + \frac{3}{31} \sqrt{5}\)

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