Question

Answer 1

\(\sqrt{4-\sqrt{7}}+\sqrt{4+\sqrt{7}}\)

\(=\dfrac{\sqrt{8-2\sqrt{7}}+\sqrt{8+2\sqrt{7}}}{\sqrt{2}}\)

\(=\dfrac{\sqrt{\left(\sqrt{7}-1\right)^2}+\sqrt{\left(\sqrt{7}+1\right)^2}}{\sqrt{2}}\)\(=\dfrac{\sqrt{7}-1+\sqrt{7}+1}{\sqrt{2}}=\dfrac{2\sqrt{7}}{\sqrt{2}}=\sqrt{14}\)

Answer 2

\(=\sqrt{\dfrac{8-2\sqrt{7}}{2}}+\sqrt{\dfrac{8+2\sqrt{7}}{2}}=\sqrt{\dfrac{\left(\sqrt{7}-1\right)^2}{2}}+\sqrt{\dfrac{\left(\sqrt{7}+1\right)^2}{2}}\\ =\dfrac{\sqrt{7}-1}{\sqrt{2}}+\dfrac{\sqrt{7}+1}{\sqrt{2}}=\dfrac{2\sqrt{7}}{\sqrt{2}}=\sqrt{14}\)