\(\sqrt{12}=\sqrt{4}⋅\sqrt{3}=2\sqrt{3}\text{.}\)

\(2\sqrt{45}=2⋅\sqrt{9}⋅\sqrt{5}=2⋅3⋅\sqrt{5}=6\sqrt{5}\text{.}\)

\(\sqrt{\frac{81}{100}}={\sqrt{81} \over \sqrt{100}}=\frac{9}{10}\text{.}\)

\(\sqrt{2\frac{8}{9}}=\sqrt{\frac{26}{9}}={\sqrt{26} \over \sqrt{9}}={\sqrt{26} \over 3}=\frac{1}{3}\sqrt{26}\text{.}\)

\(\sqrt{18}+\sqrt{8}=\sqrt{9}⋅\sqrt{2}+\sqrt{4}⋅\sqrt{2}=3\sqrt{2}+2\sqrt{2}\text{.}\)

\(3\sqrt{2}+2\sqrt{2}=5\sqrt{2}\text{.}\)

\(5\sqrt{8}-6\sqrt{32}=5⋅\sqrt{4}⋅\sqrt{2}-6⋅\sqrt{16}⋅\sqrt{2}\text{.}\)

\(5⋅2⋅\sqrt{2}-6⋅4⋅\sqrt{2}=10\sqrt{2}-24\sqrt{2}=-14\sqrt{2}\text{.}\)

\({8 \over 7\sqrt{2}}={8 \over 7\sqrt{2}}⋅{\sqrt{2} \over \sqrt{2}}={8\sqrt{2} \over 7⋅2}=\frac{4}{7}\sqrt{2}\text{.}\)

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

\(\sqrt{10\frac{2}{3}}=\sqrt{\frac{32}{3}}={\sqrt{32} \over \sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={\sqrt{96} \over 3}=\frac{1}{3}\sqrt{96}=\frac{1}{3}⋅4⋅\sqrt{6}=1\frac{1}{3}\sqrt{6}\text{.}\)

\({20\sqrt{140} \over 4\sqrt{7}}={20 \over 4}⋅{\sqrt{140} \over \sqrt{7}}=5\sqrt{20}=5⋅\sqrt{4}⋅\sqrt{5}=5⋅2⋅\sqrt{5}=10\sqrt{5}\)

\(4\sqrt{5}⋅2\sqrt{15}=8\sqrt{75}=8⋅\sqrt{25}⋅\sqrt{3}=8⋅5⋅\sqrt{3}=40\sqrt{3}\)