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

\(2\sqrt{3}+4\sqrt{3}=6\sqrt{3}\text{.}\)

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

\(-4\sqrt{27}=-4⋅\sqrt{9}⋅\sqrt{3}=-4⋅3⋅\sqrt{3}=-12\sqrt{3}\text{.}\)

\(6\sqrt{125}+2\sqrt{45}=6⋅\sqrt{25}⋅\sqrt{5}+2⋅\sqrt{9}⋅\sqrt{5}\text{.}\)

\(6⋅5⋅\sqrt{5}+2⋅3⋅\sqrt{5}=30\sqrt{5}+6\sqrt{5}=36\sqrt{5}\text{.}\)

\(\sqrt{1\frac{24}{25}}=\sqrt{\frac{49}{25}}={\sqrt{49} \over \sqrt{25}}=\frac{7}{5}=1\frac{2}{5}\text{.}\)

\({5 \over 3\sqrt{7}}={5 \over 3\sqrt{7}}⋅{\sqrt{7} \over \sqrt{7}}={5\sqrt{7} \over 3⋅7}=\frac{5}{21}\sqrt{7}\text{.}\)

\(\sqrt{1\frac{43}{49}}=\sqrt{\frac{92}{49}}={\sqrt{92} \over \sqrt{49}}={\sqrt{92} \over 7}=\frac{1}{7}\sqrt{92}=\frac{1}{7}⋅2⋅\sqrt{23}=\frac{2}{7}\sqrt{23}\text{.}\)

\(\sqrt{\frac{25}{56}}={\sqrt{25} \over \sqrt{56}}={5 \over \sqrt{56}}⋅{\sqrt{56} \over \sqrt{56}}={5\sqrt{56} \over 56}=\frac{5}{56}\sqrt{56}=\frac{5}{56}⋅2⋅\sqrt{14}=\frac{5}{28}\sqrt{14}\text{.}\)

\(\sqrt{\frac{2}{15}}={\sqrt{2} \over \sqrt{15}}⋅{\sqrt{15} \over \sqrt{15}}={\sqrt{30} \over 15}=\frac{1}{15}\sqrt{30}\text{.}\)

\({8\sqrt{84} \over 4\sqrt{3}}={8 \over 4}⋅{\sqrt{84} \over \sqrt{3}}=2\sqrt{28}=2⋅\sqrt{4}⋅\sqrt{7}=2⋅2⋅\sqrt{7}=4\sqrt{7}\)

\(2\sqrt{14}⋅5\sqrt{7}=10\sqrt{98}=10⋅\sqrt{49}⋅\sqrt{2}=10⋅7⋅\sqrt{2}=70\sqrt{2}\)

\({2 \over 4+\sqrt{5}}={2 \over 4+\sqrt{5}}⋅{4-\sqrt{5} \over 4-\sqrt{5}}\)
\(\text{}={2(4+\sqrt{5}) \over 16-5}\)
\(\text{}=\frac{2}{11}(4+\sqrt{5})\)
\(\text{}=\frac{8}{11}+\frac{2}{11}\sqrt{5}\)

\({\sqrt{5} \over \sqrt{2}+\sqrt{3}}={\sqrt{5} \over \sqrt{2}+\sqrt{3}}⋅{\sqrt{2}-\sqrt{3} \over \sqrt{2}-\sqrt{3}}\)
\(\text{}={\sqrt{5}(\sqrt{2}-\sqrt{3}) \over 2-3}\)
\(\text{}=-\sqrt{5}(\sqrt{2}-\sqrt{3})\)
\(\text{}=-\sqrt{10}+\sqrt{15}\)