Question

2 root5 + root 3 / 2 root 5 - root 3 + 2 root 5 - root 3 / 2 root 5 + 3 = a + root 15 b

Answer 1

$$\begin{gathered} \frac{2\sqrt{5}+\sqrt{3}}{2\sqrt{5}-\sqrt{3}}+\frac{2\sqrt{5}-\sqrt{3}}{2\sqrt{5}+\sqrt{3}}=\mathrm{a}+\sqrt{15}\mathrm{b} \\ \Rightarrow \frac{{\left(2\sqrt{5}+\sqrt{3}\right)}^2+{\left(2\sqrt{5}-\sqrt{3}\right)}^2}{\left(2\sqrt{5}-\sqrt{3}\right)\left(2\sqrt{5}+\sqrt{3}\right)}=\mathrm{a}+\sqrt{15}\mathrm{b} \\ \Rightarrow \frac{20+3+4\sqrt{15}+20+3-4\sqrt{15}}{20-3}=\mathrm{a}+\sqrt{15}\mathrm{b} \\ \Rightarrow \frac{46}{17}=\mathrm{a}+\sqrt{15}\mathrm{b} \\ \Rightarrow \mathrm{a}=\frac{46}{17}\mathrm{and}\mathrm{b}=0 \end{gathered}$$