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

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

\(\sqrt{20} = \sqrt{4} ⋅ \sqrt{5} = 2 \sqrt{5} \text{.}\)

\(7 \sqrt{80} = 7 ⋅ \sqrt{16} ⋅ \sqrt{5} = 7 ⋅ 4 ⋅ \sqrt{5} = 28 \sqrt{5} \text{.}\)

\(5 \sqrt{500} + 2 \sqrt{45} = 5 ⋅ \sqrt{100} ⋅ \sqrt{5} + 2 ⋅ \sqrt{9} ⋅ \sqrt{5} \text{.}\)

\(5 ⋅ 10 ⋅ \sqrt{5} + 2 ⋅ 3 ⋅ \sqrt{5} = 50 \sqrt{5} + 6 \sqrt{5} = 56 \sqrt{5} \text{.}\)

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

\({9 \over 8 \sqrt{5}} = {9 \over 8 \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {9 \sqrt{5} \over 8 ⋅ 5} = \frac{9}{40} \sqrt{5} \text{.}\)

\(\sqrt{3\frac{17}{25}} = \sqrt{\frac{92}{25}} = {\sqrt{92} \over \sqrt{25}} = {\sqrt{92} \over 5} = \frac{1}{5} \sqrt{92} = \frac{1}{5} ⋅ 2 ⋅ \sqrt{23} = \frac{2}{5} \sqrt{23} \text{.}\)

\(\sqrt{2\frac{19}{31}} = \sqrt{\frac{81}{31}} = {\sqrt{81} \over \sqrt{31}} = {9 \over \sqrt{31}} ⋅ {\sqrt{31} \over \sqrt{31}} = {9 \sqrt{31} \over 31} = \frac{9}{31} \sqrt{31} \text{.}\)

\(\sqrt{\frac{8}{75}} = {\sqrt{8} \over \sqrt{75}} ⋅ {\sqrt{75} \over \sqrt{75}} = {\sqrt{600} \over 75} = \frac{1}{75} \sqrt{600} = \frac{1}{75} ⋅ 10 ⋅ \sqrt{6} = \frac{2}{15} \sqrt{6} \text{.}\)

\({72 \sqrt{100} \over 8 \sqrt{5}} = {72 \over 8} ⋅ {\sqrt{100} \over \sqrt{5}} = 9 \sqrt{20} = 9 ⋅ \sqrt{4} ⋅ \sqrt{5} = 9 ⋅ 2 ⋅ \sqrt{5} = 18 \sqrt{5}\)

\(4 \sqrt{15} ⋅ 3 \sqrt{3} = 12 \sqrt{45} = 12 ⋅ \sqrt{9} ⋅ \sqrt{5} = 12 ⋅ 3 ⋅ \sqrt{5} = 36 \sqrt{5}\)