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

\(-5\sqrt{8}=-5⋅\sqrt{4}⋅\sqrt{2}=-5⋅2⋅\sqrt{2}=-10\sqrt{2}\text{.}\)

\(\sqrt{2\frac{14}{25}}=\sqrt{\frac{64}{25}}={\sqrt{64} \over \sqrt{25}}=\frac{8}{5}=1\frac{3}{5}\text{.}\)

\({24\sqrt{320} \over 3\sqrt{8}}={24 \over 3}⋅{\sqrt{320} \over \sqrt{8}}=8\sqrt{40}=8⋅\sqrt{4}⋅\sqrt{10}=8⋅2⋅\sqrt{10}=16\sqrt{10}\)

\(\sqrt{\frac{48}{49}}={\sqrt{48} \over \sqrt{49}}={\sqrt{48} \over 7}=\frac{1}{7}\sqrt{48}=\frac{1}{7}⋅4⋅\sqrt{3}=\frac{4}{7}\sqrt{3}\text{.}\)

\(\sqrt{500}+\sqrt{45}=\sqrt{100}⋅\sqrt{5}+\sqrt{9}⋅\sqrt{5}=10\sqrt{5}+3\sqrt{5}\text{.}\)

\(10\sqrt{5}+3\sqrt{5}=13\sqrt{5}\text{.}\)

\(2\sqrt{500}-4\sqrt{125}=2⋅\sqrt{100}⋅\sqrt{5}-4⋅\sqrt{25}⋅\sqrt{5}\text{.}\)

\(2⋅10⋅\sqrt{5}-4⋅5⋅\sqrt{5}=20\sqrt{5}-20\sqrt{5}=0\sqrt{5}\text{.}\)

\({5 \over 8\sqrt{2}}={5 \over 8\sqrt{2}}⋅{\sqrt{2} \over \sqrt{2}}={5\sqrt{2} \over 8⋅2}=\frac{5}{16}\sqrt{2}\text{.}\)

\(\sqrt{\frac{1}{84}}={\sqrt{1} \over \sqrt{84}}={1 \over \sqrt{84}}⋅{\sqrt{84} \over \sqrt{84}}={\sqrt{84} \over 84}=\frac{1}{84}\sqrt{84}=\frac{1}{84}⋅2⋅\sqrt{21}=\frac{1}{42}\sqrt{21}\text{.}\)

\(\sqrt{\frac{2}{75}}={\sqrt{2} \over \sqrt{75}}⋅{\sqrt{75} \over \sqrt{75}}={\sqrt{150} \over 75}=\frac{1}{75}\sqrt{150}=\frac{1}{75}⋅5⋅\sqrt{6}=\frac{1}{15}\sqrt{6}\text{.}\)