\(\sqrt{45}+\sqrt{125}=\sqrt{9}⋅\sqrt{5}+\sqrt{25}⋅\sqrt{5}=3\sqrt{5}+5\sqrt{5}\text{.}\)

\(3\sqrt{5}+5\sqrt{5}=8\sqrt{5}\text{.}\)

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

\(4\sqrt{175}=4⋅\sqrt{25}⋅\sqrt{7}=4⋅5⋅\sqrt{7}=20\sqrt{7}\text{.}\)

\(4\sqrt{50}+2\sqrt{200}=4⋅\sqrt{25}⋅\sqrt{2}+2⋅\sqrt{100}⋅\sqrt{2}\text{.}\)

\(4⋅5⋅\sqrt{2}+2⋅10⋅\sqrt{2}=20\sqrt{2}+20\sqrt{2}=40\sqrt{2}\text{.}\)

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

\({4 \over 9\sqrt{3}}={4 \over 9\sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={4\sqrt{3} \over 9⋅3}=\frac{4}{27}\sqrt{3}\text{.}\)

\(\sqrt{\frac{56}{81}}={\sqrt{56} \over \sqrt{81}}={\sqrt{56} \over 9}=\frac{1}{9}\sqrt{56}=\frac{1}{9}⋅2⋅\sqrt{14}=\frac{2}{9}\sqrt{14}\text{.}\)

\(\sqrt{\frac{16}{45}}={\sqrt{16} \over \sqrt{45}}={4 \over \sqrt{45}}⋅{\sqrt{45} \over \sqrt{45}}={4\sqrt{45} \over 45}=\frac{4}{45}\sqrt{45}=\frac{4}{45}⋅3⋅\sqrt{5}=\frac{4}{15}\sqrt{5}\text{.}\)

\(\sqrt{1\frac{2}{3}}=\sqrt{\frac{5}{3}}={\sqrt{5} \over \sqrt{3}}⋅{\sqrt{3} \over \sqrt{3}}={\sqrt{15} \over 3}=\frac{1}{3}\sqrt{15}\text{.}\)

\({45\sqrt{72} \over 5\sqrt{3}}={45 \over 5}⋅{\sqrt{72} \over \sqrt{3}}=9\sqrt{24}=9⋅\sqrt{4}⋅\sqrt{6}=9⋅2⋅\sqrt{6}=18\sqrt{6}\)

\(2\sqrt{2}⋅3\sqrt{14}=6\sqrt{28}=6⋅\sqrt{4}⋅\sqrt{7}=6⋅2⋅\sqrt{7}=12\sqrt{7}\)

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

\({4\sqrt{2} \over \sqrt{5}-\sqrt{6}}={4\sqrt{2} \over \sqrt{5}-\sqrt{6}}⋅{\sqrt{5}+\sqrt{6} \over \sqrt{5}+\sqrt{6}}\)
\(\text{}={4\sqrt{2}(\sqrt{5}+\sqrt{6}) \over 5-6}\)
\(\text{}=-4\sqrt{2}(\sqrt{5}+\sqrt{6})\)
\(\text{}=-4\sqrt{10}-4\sqrt{12}\)
\(\text{}=-4\sqrt{10}-8\sqrt{3}\)