\(\sqrt{18}=\sqrt{9}⋅\sqrt{2}=3\sqrt{2}\text{.}\)

\(6\sqrt{500}=6⋅\sqrt{100}⋅\sqrt{5}=6⋅10⋅\sqrt{5}=60\sqrt{5}\text{.}\)

\(\sqrt{\frac{4}{25}}={\sqrt{4} \over \sqrt{25}}=\frac{2}{5}\text{.}\)

\({24\sqrt{200} \over 3\sqrt{10}}={24 \over 3}⋅{\sqrt{200} \over \sqrt{10}}=8\sqrt{20}=8⋅\sqrt{4}⋅\sqrt{5}=8⋅2⋅\sqrt{5}=16\sqrt{5}\)

\(\sqrt{1\frac{19}{49}}=\sqrt{\frac{68}{49}}={\sqrt{68} \over \sqrt{49}}={\sqrt{68} \over 7}=\frac{1}{7}\sqrt{68}=\frac{1}{7}⋅2⋅\sqrt{17}=\frac{2}{7}\sqrt{17}\text{.}\)