Question

Simplify ( 3 + root 6) / (5 root 3 - 2 root 12 - root 30 + root 50)

Answer 1

Dear student
$$\begin{gathered} \frac{3+\sqrt{6}}{5\sqrt{3}-2\sqrt{12}-\sqrt{30}+\sqrt{50}} \\ =\frac{3+\sqrt{6}}{5\sqrt{3}-4\sqrt{3}-\sqrt{30}+5\sqrt{2}} \\ =\frac{3+\sqrt{6}}{\sqrt{3}-\sqrt{30}+5\sqrt{2}} \\ =\frac{3+\sqrt{6}}{\sqrt{3}-\sqrt{2}\sqrt{15}+5\sqrt{2}} \\ =\frac{3+\sqrt{6}}{\sqrt{3}-\sqrt{2}\left(\sqrt{15}-5\right)} \\ =\frac{3+\sqrt{6}}{\sqrt{3}-\sqrt{2}\left(\sqrt{15}-5\right)}\times \frac{\sqrt{3}+\sqrt{2}\left(\sqrt{15}-5\right)}{\sqrt{3}+\sqrt{2}\left(\sqrt{15}-5\right)} \\ =\frac{3\sqrt{3}+3\sqrt{2}\left(\sqrt{15}-5\right)+3\sqrt{2}+2\sqrt{3}\left(\sqrt{15}-5\right)}{3-2{\left(\sqrt{15}-5\right)}^2} \\ =\frac{\sqrt{3}\left[3+2\left(\sqrt{15}-5\right)\right]+\sqrt{2}\left[3+2\left(\sqrt{15}-5\right)\right]}{3-2\left(15+25-10\sqrt{15}\right)} \\ =\frac{\sqrt{3}\left[3+2\left(\sqrt{15}-5\right)\right]+\sqrt{2}\left[3+2\left(\sqrt{15}-5\right)\right]}{3-30-50+20\sqrt{15}} \\ =\frac{\sqrt{3}\left[3+2\left(\sqrt{15}-5\right)\right]+\sqrt{2}\left[3+2\left(\sqrt{15}-5\right)\right]}{-77+20\sqrt{15}} \\ =\frac{\sqrt{3}\left[3+2\left(\sqrt{15}-5\right)\right]+\sqrt{2}\left[3+2\left(\sqrt{15}-5\right)\right]}{20\sqrt{15}-77}\times \frac{20\sqrt{15}+77}{20\sqrt{15}+77} \\ =\frac{\left[\sqrt{3}\left[3+2\left(\sqrt{15}-5\right)\right]+\sqrt{2}\left[3+2\left(\sqrt{15}-5\right)\right]\right]\left[20\sqrt{15}+77\right]}{6000-5929} \\ =\frac{\left[\sqrt{3}\left[3+2\left(\sqrt{15}-5\right)\right]+\sqrt{2}\left[3+2\left(\sqrt{15}-5\right)\right]\right]\left[20\sqrt{15}+77\right]}{-71} \end{gathered}$$
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