Question

If $\sqrt{5+2\sqrt{6}}=A+\sqrt{3},\mathrm{then}A=\_\_\_\_\_\_\_\_\_.$

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \sqrt{5+2\sqrt{6}}=A+\sqrt{3} \\ \mathrm{Now}, \\ \sqrt{5+2\sqrt{6}}=A+\sqrt{3} \\ \Rightarrow \sqrt{2+3+2\sqrt{6}}=A+\sqrt{3} \\ \Rightarrow \sqrt{{\left(\sqrt{2}\right)}^2+{\left(\sqrt{3}\right)}^2+2\sqrt{6}}=A+\sqrt{3} \\ \Rightarrow \sqrt{{\left(\sqrt{2}+\sqrt{3}\right)}^2}=A+\sqrt{3}\left(\mathrm{Using}\mathrm{the}\mathrm{identity}:\hspace{0.17em}{\left(a+b\right)}^2=a^2+b^2+2ab\right) \\ \Rightarrow \sqrt{2}+\sqrt{3}=A+\sqrt{3} \\ \Rightarrow A=\sqrt{2} \end{gathered}$$


Hence, if $\sqrt{5+2\sqrt{6}}=A+\sqrt{3},\mathrm{then}A=\underline{\sqrt{2}}.$