\(\sqrt{125} + \sqrt{80} = \sqrt{25} ⋅ \sqrt{5} + \sqrt{16} ⋅ \sqrt{5} = 5 \sqrt{5} + 4 \sqrt{5} \text{.}\)

\(5 \sqrt{5} + 4 \sqrt{5} = 9 \sqrt{5} \text{.}\)

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

\(6 \sqrt{8} = 6 ⋅ \sqrt{4} ⋅ \sqrt{2} = 6 ⋅ 2 ⋅ \sqrt{2} = 12 \sqrt{2} \text{.}\)

\(6 \sqrt{300} - 5 \sqrt{48} = 6 ⋅ \sqrt{100} ⋅ \sqrt{3} - 5 ⋅ \sqrt{16} ⋅ \sqrt{3} \text{.}\)

\(6 ⋅ 10 ⋅ \sqrt{3} - 5 ⋅ 4 ⋅ \sqrt{3} = 60 \sqrt{3} - 20 \sqrt{3} = 40 \sqrt{3} \text{.}\)

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

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

\(\sqrt{\frac{79}{81}} = {\sqrt{79} \over \sqrt{81}} = {\sqrt{79} \over 9} = \frac{1}{9} \sqrt{79} \text{.}\)

\(\sqrt{1\frac{1}{63}} = \sqrt{\frac{64}{63}} = {\sqrt{64} \over \sqrt{63}} = {8 \over \sqrt{63}} ⋅ {\sqrt{63} \over \sqrt{63}} = {8 \sqrt{63} \over 63} = \frac{8}{63} \sqrt{63} = \frac{8}{63} ⋅ 3 ⋅ \sqrt{7} = \frac{8}{21} \sqrt{7} \text{.}\)

\(\sqrt{\frac{3}{17}} = {\sqrt{3} \over \sqrt{17}} ⋅ {\sqrt{17} \over \sqrt{17}} = {\sqrt{51} \over 17} = \frac{1}{17} \sqrt{51} \text{.}\)

\({15 \sqrt{120} \over 5 \sqrt{10}} = {15 \over 5} ⋅ {\sqrt{120} \over \sqrt{10}} = 3 \sqrt{12} = 3 ⋅ \sqrt{4} ⋅ \sqrt{3} = 3 ⋅ 2 ⋅ \sqrt{3} = 6 \sqrt{3}\)

\(3 \sqrt{15} ⋅ 2 \sqrt{5} = 6 \sqrt{75} = 6 ⋅ \sqrt{25} ⋅ \sqrt{3} = 6 ⋅ 5 ⋅ \sqrt{3} = 30 \sqrt{3}\)