\(\sqrt{8}+\sqrt{200}=\sqrt{4}⋅\sqrt{2}+\sqrt{100}⋅\sqrt{2}=2\sqrt{2}+10\sqrt{2}\text{.}\)

\(2\sqrt{2}+10\sqrt{2}=12\sqrt{2}\text{.}\)

\(\sqrt{200}=\sqrt{100}⋅\sqrt{2}=10\sqrt{2}\text{.}\)

\(-2\sqrt{48}=-2⋅\sqrt{16}⋅\sqrt{3}=-2⋅4⋅\sqrt{3}=-8\sqrt{3}\text{.}\)

\(2\sqrt{48}+6\sqrt{12}=2⋅\sqrt{16}⋅\sqrt{3}+6⋅\sqrt{4}⋅\sqrt{3}\text{.}\)

\(2⋅4⋅\sqrt{3}+6⋅2⋅\sqrt{3}=8\sqrt{3}+12\sqrt{3}=20\sqrt{3}\text{.}\)

\(\sqrt{1\frac{15}{49}}=\sqrt{\frac{64}{49}}={\sqrt{64} \over \sqrt{49}}=\frac{8}{7}=1\frac{1}{7}\text{.}\)

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

\(\sqrt{18\frac{3}{4}}=\sqrt{\frac{75}{4}}={\sqrt{75} \over \sqrt{4}}={\sqrt{75} \over 2}=\frac{1}{2}\sqrt{75}=\frac{1}{2}⋅5⋅\sqrt{3}=2\frac{1}{2}\sqrt{3}\text{.}\)

\(\sqrt{\frac{25}{44}}={\sqrt{25} \over \sqrt{44}}={5 \over \sqrt{44}}⋅{\sqrt{44} \over \sqrt{44}}={5\sqrt{44} \over 44}=\frac{5}{44}\sqrt{44}=\frac{5}{44}⋅2⋅\sqrt{11}=\frac{5}{22}\sqrt{11}\text{.}\)

\(\sqrt{14\frac{1}{2}}=\sqrt{\frac{29}{2}}={\sqrt{29} \over \sqrt{2}}⋅{\sqrt{2} \over \sqrt{2}}={\sqrt{58} \over 2}=\frac{1}{2}\sqrt{58}\text{.}\)

\({40\sqrt{280} \over 5\sqrt{7}}={40 \over 5}⋅{\sqrt{280} \over \sqrt{7}}=8\sqrt{40}=8⋅\sqrt{4}⋅\sqrt{10}=8⋅2⋅\sqrt{10}=16\sqrt{10}\)

\(2\sqrt{14}⋅3\sqrt{7}=6\sqrt{98}=6⋅\sqrt{49}⋅\sqrt{2}=6⋅7⋅\sqrt{2}=42\sqrt{2}\)

\({-3 \over 5+\sqrt{2}}={-3 \over 5+\sqrt{2}}⋅{5-\sqrt{2} \over 5-\sqrt{2}}\)
\(\text{}={-3(5+\sqrt{2}) \over 25-2}\)
\(\text{}=-\frac{3}{23}(5+\sqrt{2})\)
\(\text{}=-\frac{15}{23}-\frac{3}{23}\sqrt{2}\)

\({4\sqrt{2} \over \sqrt{5}-\sqrt{3}}={4\sqrt{2} \over \sqrt{5}-\sqrt{3}}⋅{\sqrt{5}+\sqrt{3} \over \sqrt{5}+\sqrt{3}}\)
\(\text{}={4\sqrt{2}(\sqrt{5}+\sqrt{3}) \over 5-3}\)
\(\text{}=2\sqrt{2}(\sqrt{5}+\sqrt{3})\)
\(\text{}=2\sqrt{10}+2\sqrt{6}\)