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1. If N = $\frac{\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}}{\sqrt{\sqrt{5} + 1}} -$ $\sqrt{3 - 2 \sqrt{2}},$ then the value of N is = ?

A. $2 \sqrt{2} - 1$

B. 3

C. 1

D. 2

Solution:

\begin{aligned} Let X = \frac{\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}}{\sqrt{\sqrt{5} + 1}} \\ Then, \\ X^{2} = \frac{{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2})}^{2}}{{(\sqrt{\sqrt{5} + 1})}^{2}} \end{aligned}

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

\begin{aligned} X^{2} = \frac{2 \sqrt{5} + 2 \sqrt{{(\sqrt{5})}^{2} - {(2)}^{2}}}{\sqrt{5} + 1} \\ X^{2} = \frac{2 \sqrt{5} + 2}{\sqrt{5} + 1} \\ X^{2} = \frac{2 (\sqrt{5} + 1)}{(\sqrt{5} + 1)} \\ X^{2} = 2 \\ X = \sqrt{2} \\ ∴ N = \sqrt{2} - \sqrt{3 - 2 \sqrt{2}} \\ N = \sqrt{2} - \sqrt{{(\sqrt{2})}^{2} + 1^{2} - 2 \times \sqrt{2} \times} 1 \\ N = \sqrt{2} - \sqrt{{(\sqrt{2} - 1)}^{2}} \\ N = \sqrt{2} - (\sqrt{2} - 1) \\ N = 1 \end{aligned}

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