\(\sqrt{500}+\sqrt{125}=\sqrt{100}⋅\sqrt{5}+\sqrt{25}⋅\sqrt{5}=10\sqrt{5}+5\sqrt{5}\text{.}\)

\(10\sqrt{5}+5\sqrt{5}=15\sqrt{5}\text{.}\)

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

\(4\sqrt{50}=4⋅\sqrt{25}⋅\sqrt{2}=4⋅5⋅\sqrt{2}=20\sqrt{2}\text{.}\)

\(7\sqrt{300}+6\sqrt{75}=7⋅\sqrt{100}⋅\sqrt{3}+6⋅\sqrt{25}⋅\sqrt{3}\text{.}\)

\(7⋅10⋅\sqrt{3}+6⋅5⋅\sqrt{3}=70\sqrt{3}+30\sqrt{3}=100\sqrt{3}\text{.}\)

\(\sqrt{1\frac{7}{9}}=\sqrt{\frac{16}{9}}={\sqrt{16} \over \sqrt{9}}=\frac{4}{3}=1\frac{1}{3}\text{.}\)

\({8 \over 3\sqrt{2}}={8 \over 3\sqrt{2}}⋅{\sqrt{2} \over \sqrt{2}}={8\sqrt{2} \over 3⋅2}=1\frac{1}{3}\sqrt{2}\text{.}\)

\(\sqrt{\frac{24}{25}}={\sqrt{24} \over \sqrt{25}}={\sqrt{24} \over 5}=\frac{1}{5}\sqrt{24}=\frac{1}{5}⋅2⋅\sqrt{6}=\frac{2}{5}\sqrt{6}\text{.}\)

\(\sqrt{\frac{1}{96}}={\sqrt{1} \over \sqrt{96}}={1 \over \sqrt{96}}⋅{\sqrt{96} \over \sqrt{96}}={\sqrt{96} \over 96}=\frac{1}{96}\sqrt{96}=\frac{1}{96}⋅4⋅\sqrt{6}=\frac{1}{24}\sqrt{6}\text{.}\)

\(\sqrt{2\frac{1}{2}}=\sqrt{\frac{5}{2}}={\sqrt{5} \over \sqrt{2}}⋅{\sqrt{2} \over \sqrt{2}}={\sqrt{10} \over 2}=\frac{1}{2}\sqrt{10}\text{.}\)

\({45\sqrt{160} \over 5\sqrt{8}}={45 \over 5}⋅{\sqrt{160} \over \sqrt{8}}=9\sqrt{20}=9⋅\sqrt{4}⋅\sqrt{5}=9⋅2⋅\sqrt{5}=18\sqrt{5}\)

\(2\sqrt{6}⋅5\sqrt{10}=10\sqrt{60}=10⋅\sqrt{4}⋅\sqrt{15}=10⋅2⋅\sqrt{15}=20\sqrt{15}\)