Question

$\sqrt{7+2\sqrt{6}}-\sqrt{7-2\sqrt{6}}=\_\_\_\_\_\_\_\_\_\_.$

Answer 1

$$\begin{gathered} \sqrt{7+2\sqrt{6}} \\ =\sqrt{6+1+2\sqrt{6}} \\ =\sqrt{{\left(\sqrt{6}\right)}^2+{\left(1\right)}^2+2\sqrt{6}} \\ =\sqrt{{\left(\sqrt{6}+1\right)}^2}\left(\mathrm{Using}\mathrm{the}\mathrm{identity}:\hspace{0.17em}{\left(a+b\right)}^2=a^2+b^2+2ab\right) \\ =\sqrt{6}+1....\left(1\right) \\ \sqrt{7-2\sqrt{6}} \\ =\sqrt{6+1-2\sqrt{6}} \\ =\sqrt{{\left(\sqrt{6}\right)}^2+{\left(1\right)}^2-2\sqrt{6}} \\ =\sqrt{{\left(\sqrt{6}-1\right)}^2}\left(\mathrm{Using}\mathrm{the}\mathrm{identity}:\hspace{0.17em}{\left(a-b\right)}^2=a^2+b^2-2ab\right) \\ =\sqrt{6}-1....\left(2\right) \\ \mathrm{From}\left(1\right)\mathrm{and}\left(2\right) \\ \sqrt{7+2\sqrt{6}}-\sqrt{7-2\sqrt{6}}=\left(\sqrt{6}+1\right)-\left(\sqrt{6}-1\right) \\ =\sqrt{6}+1-\sqrt{6}+1 \\ =2 \end{gathered}$$


Hence, $\sqrt{7+2\sqrt{6}}-\sqrt{7-2\sqrt{6}}=\underline{2}.$