Question

whole square root(root5+2) + whole square root (root 5 -2 )/ whole squareroot (root5 + 1) - whole square root (3-2root2)

Answer 1

Dear student
$$\begin{gathered} \frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}-\sqrt{3-2\sqrt{2}}} \\ \mathrm{Now},\sqrt{3-2\sqrt{2}}=\sqrt{{\left(\sqrt{2}\right)}^2+1^2-2\sqrt{2}}=\sqrt{{\left(\sqrt{2}-1\right)}^2}=\sqrt{2}-1 \\ \mathrm{So}, \\ \frac{{\displaystyle \sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}}{{\displaystyle \sqrt{\sqrt{5}+1}-\sqrt{3-2\sqrt{2}}}} \\ =\frac{{\displaystyle \sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}}{{\displaystyle \sqrt{\sqrt{5}+1}-\sqrt{2}-1}} \\ =\frac{{\displaystyle \sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}}{{\displaystyle \sqrt{\sqrt{5}+1}-\sqrt{2}-1}}\times \frac{\sqrt{\sqrt{5}+2}{\displaystyle -}\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+2}{\displaystyle -}\sqrt{\sqrt{5}-2}} \\ =\frac{\sqrt{5}+2-\sqrt{5}+2}{\left(\sqrt{\sqrt{5}+1}-\sqrt{2}-1\right){\displaystyle \left(\sqrt{\sqrt{5}+2}-\sqrt{\sqrt{5}-2}\right)}} \\ =\frac{4}{\left(\sqrt{\sqrt{5}+1}-\sqrt{2}-1\right)\left(\sqrt{\sqrt{5}+2}-\sqrt{\sqrt{5}-2}\right)} \end{gathered}$$
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