Question

\(\frac{\sqrt{\text{√}5+2}+\sqrt{\text{√}5-2}}{\sqrt{\text{√}5+1}}\)\(-\sqrt{6-2\text{√}8}\)

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Answer 1

\(=\sqrt{\frac{\sqrt{5}+2}{\sqrt{5}+1}}+\sqrt{\frac{\sqrt{5}-2}{\sqrt{5}+1}}-\sqrt{3-2\sqrt{3}.\sqrt{2}+2}\)

\(=\sqrt{\frac{3+\sqrt{5}}{4}}+\sqrt{\frac{7-3\sqrt{5}}{4}}-\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}\)

\(=\sqrt{\frac{6+2\sqrt{5}}{8}}+\sqrt{\frac{14-2.3\sqrt{5}}{8}}-\left(\sqrt{3}-\sqrt{2}\right)\)

\(=\sqrt{\frac{\left(\sqrt{5}+1\right)^2}{8}}+\sqrt{\frac{\left(3-\sqrt{5}\right)^2}{8}}-\left(\sqrt{3}-\sqrt{2}\right)\)

\(=\frac{\sqrt{5}+1+3-\sqrt{5}}{2\sqrt{2}}-\sqrt{3}+\sqrt{2}\)=\(\sqrt{2}-\sqrt{3}+\sqrt{2}=-\sqrt{3}\)