\(\sqrt{75} + \sqrt{27} = \sqrt{25} ⋅ \sqrt{3} + \sqrt{9} ⋅ \sqrt{3} = 5 \sqrt{3} + 3 \sqrt{3} \text{.}\)

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

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

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

\(6 \sqrt{500} - 7 \sqrt{45} = 6 ⋅ \sqrt{100} ⋅ \sqrt{5} - 7 ⋅ \sqrt{9} ⋅ \sqrt{5} \text{.}\)

\(6 ⋅ 10 ⋅ \sqrt{5} - 7 ⋅ 3 ⋅ \sqrt{5} = 60 \sqrt{5} - 21 \sqrt{5} = 39 \sqrt{5} \text{.}\)

\(\sqrt{\frac{1}{36}} = {\sqrt{1} \over \sqrt{36}} = \frac{1}{6} \text{.}\)

\({5 \over 7 \sqrt{5}} = {5 \over 7 \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {5 \sqrt{5} \over 7 ⋅ 5} = \frac{1}{7} \sqrt{5} \text{.}\)

\(\sqrt{\frac{11}{16}} = {\sqrt{11} \over \sqrt{16}} = {\sqrt{11} \over 4} = \frac{1}{4} \sqrt{11} \text{.}\)

\(\sqrt{\frac{1}{32}} = {\sqrt{1} \over \sqrt{32}} = {1 \over \sqrt{32}} ⋅ {\sqrt{32} \over \sqrt{32}} = {\sqrt{32} \over 32} = \frac{1}{32} \sqrt{32} = \frac{1}{32} ⋅ 4 ⋅ \sqrt{2} = \frac{1}{8} \sqrt{2} \text{.}\)

\(\sqrt{8\frac{1}{2}} = \sqrt{\frac{17}{2}} = {\sqrt{17} \over \sqrt{2}} ⋅ {\sqrt{2} \over \sqrt{2}} = {\sqrt{34} \over 2} = \frac{1}{2} \sqrt{34} \text{.}\)

\({9 \sqrt{120} \over \sqrt{3}} = 9 ⋅ {\sqrt{120} \over \sqrt{3}} = 9 \sqrt{40} = 9 ⋅ \sqrt{4} ⋅ \sqrt{10} = 9 ⋅ 2 ⋅ \sqrt{10} = 18 \sqrt{10}\)

\(2 \sqrt{10} ⋅ 3 \sqrt{6} = 6 \sqrt{60} = 6 ⋅ \sqrt{4} ⋅ \sqrt{15} = 6 ⋅ 2 ⋅ \sqrt{15} = 12 \sqrt{15}\)