Question

1) $\sqrt{6-\sqrt{11}}$ + $\sqrt{6+\sqrt{11}}$

2) $\sqrt{4-\sqrt{15}}$ - $\sqrt{4+\sqrt{15}}$

Answer 1

1: \(\sqrt{6-\sqrt{11}}+\sqrt{6+\sqrt{11}}\)

\(=\dfrac{\sqrt{12-2\sqrt{11}}+\sqrt{12+2\sqrt{11}}}{\sqrt{2}}\)

\(=\dfrac{\sqrt{11}-1+\sqrt{11}+1}{\sqrt{2}}=\dfrac{2\sqrt{11}}{\sqrt{2}}=\sqrt{22}\)

2: \(\sqrt{4-\sqrt{15}}-\sqrt{4+\sqrt{15}}\)

\(=\dfrac{\sqrt{8-2\sqrt{15}}-\sqrt{8+2\sqrt{15}}}{\sqrt{2}}\)

\(=\dfrac{\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{5}+\sqrt{3}\right)^2}}{\sqrt{2}}\)

\(=\dfrac{\sqrt{5}-\sqrt{3}-\sqrt{5}-\sqrt{3}}{\sqrt{2}}=-\dfrac{2\sqrt{3}}{\sqrt{2}}=-\sqrt{6}\)