\(\sqrt{500} = \sqrt{100} ⋅ \sqrt{5} = 10 \sqrt{5} \text{.}\)

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

\(\sqrt{\frac{1}{100}} = {\sqrt{1} \over \sqrt{100}} = \frac{1}{10} \text{.}\)

\({36 \sqrt{120} \over 4 \sqrt{6}} = {36 \over 4} ⋅ {\sqrt{120} \over \sqrt{6}} = 9 \sqrt{20} = 9 ⋅ \sqrt{4} ⋅ \sqrt{5} = 9 ⋅ 2 ⋅ \sqrt{5} = 18 \sqrt{5}\)

\(\sqrt{5\frac{13}{16}} = \sqrt{\frac{93}{16}} = {\sqrt{93} \over \sqrt{16}} = {\sqrt{93} \over 4} = \frac{1}{4} \sqrt{93} \text{.}\)

\(\sqrt{20} + \sqrt{125} = \sqrt{4} ⋅ \sqrt{5} + \sqrt{25} ⋅ \sqrt{5} = 2 \sqrt{5} + 5 \sqrt{5} \text{.}\)

\(2 \sqrt{5} + 5 \sqrt{5} = 7 \sqrt{5} \text{.}\)

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

\(7 ⋅ 4 ⋅ \sqrt{5} - 4 ⋅ 2 ⋅ \sqrt{5} = 28 \sqrt{5} - 8 \sqrt{5} = 20 \sqrt{5} \text{.}\)

\({8 \over 7 \sqrt{7}} = {8 \over 7 \sqrt{7}} ⋅ {\sqrt{7} \over \sqrt{7}} = {8 \sqrt{7} \over 7 ⋅ 7} = \frac{8}{49} \sqrt{7} \text{.}\)

\(\sqrt{3\frac{4}{15}} = \sqrt{\frac{49}{15}} = {\sqrt{49} \over \sqrt{15}} = {7 \over \sqrt{15}} ⋅ {\sqrt{15} \over \sqrt{15}} = {7 \sqrt{15} \over 15} = \frac{7}{15} \sqrt{15} \text{.}\)

\(\sqrt{\frac{3}{13}} = {\sqrt{3} \over \sqrt{13}} ⋅ {\sqrt{13} \over \sqrt{13}} = {\sqrt{39} \over 13} = \frac{1}{13} \sqrt{39} \text{.}\)

\(2 \sqrt{2} ⋅ 5 \sqrt{14} = 10 \sqrt{28} = 10 ⋅ \sqrt{4} ⋅ \sqrt{7} = 10 ⋅ 2 ⋅ \sqrt{7} = 20 \sqrt{7}\)