\(\sqrt{80} + \sqrt{500} = \sqrt{16} ⋅ \sqrt{5} + \sqrt{100} ⋅ \sqrt{5} = 4 \sqrt{5} + 10 \sqrt{5} \text{.}\)

\(4 \sqrt{5} + 10 \sqrt{5} = 14 \sqrt{5} \text{.}\)

\(\sqrt{125} = \sqrt{25} ⋅ \sqrt{5} = 5 \sqrt{5} \text{.}\)

\(-5 \sqrt{63} = -5 ⋅ \sqrt{9} ⋅ \sqrt{7} = -5 ⋅ 3 ⋅ \sqrt{7} = -15 \sqrt{7} \text{.}\)

\(2 \sqrt{12} - 5 \sqrt{300} = 2 ⋅ \sqrt{4} ⋅ \sqrt{3} - 5 ⋅ \sqrt{100} ⋅ \sqrt{3} \text{.}\)

\(2 ⋅ 2 ⋅ \sqrt{3} - 5 ⋅ 10 ⋅ \sqrt{3} = 4 \sqrt{3} - 50 \sqrt{3} = -46 \sqrt{3} \text{.}\)

\(\sqrt{2\frac{1}{4}} = \sqrt{\frac{9}{4}} = {\sqrt{9} \over \sqrt{4}} = \frac{3}{2} = 1\frac{1}{2} \text{.}\)

\({5 \over 9 \sqrt{7}} = {5 \over 9 \sqrt{7}} ⋅ {\sqrt{7} \over \sqrt{7}} = {5 \sqrt{7} \over 9 ⋅ 7} = \frac{5}{63} \sqrt{7} \text{.}\)

\(\sqrt{\frac{76}{81}} = {\sqrt{76} \over \sqrt{81}} = {\sqrt{76} \over 9} = \frac{1}{9} \sqrt{76} = \frac{1}{9} ⋅ 2 ⋅ \sqrt{19} = \frac{2}{9} \sqrt{19} \text{.}\)

\(\sqrt{2\frac{15}{17}} = \sqrt{\frac{49}{17}} = {\sqrt{49} \over \sqrt{17}} = {7 \over \sqrt{17}} ⋅ {\sqrt{17} \over \sqrt{17}} = {7 \sqrt{17} \over 17} = \frac{7}{17} \sqrt{17} \text{.}\)

\(\sqrt{\frac{7}{13}} = {\sqrt{7} \over \sqrt{13}} ⋅ {\sqrt{13} \over \sqrt{13}} = {\sqrt{91} \over 13} = \frac{1}{13} \sqrt{91} \text{.}\)

\({28 \sqrt{168} \over 7 \sqrt{7}} = {28 \over 7} ⋅ {\sqrt{168} \over \sqrt{7}} = 4 \sqrt{24} = 4 ⋅ \sqrt{4} ⋅ \sqrt{6} = 4 ⋅ 2 ⋅ \sqrt{6} = 8 \sqrt{6}\)

\(5 \sqrt{2} ⋅ 3 \sqrt{10} = 15 \sqrt{20} = 15 ⋅ \sqrt{4} ⋅ \sqrt{5} = 15 ⋅ 2 ⋅ \sqrt{5} = 30 \sqrt{5}\)