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

Concept & Formula For complex nested radical fractions, isolate the numerator and denominator to simplify them independently. Squaring a sum of conjugate surds effectively eliminates the innermost radicals. $$ (x + y)^2 = x^2 + y^2 + 2xy $$ Step-by-Step Solution * Simplify the numerator: Let $x = \sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}$ * Square both sides: $$ x^2 = (\sqrt{5} + 2) + (\sqrt{5} - 2) + 2\sqrt{(\sqrt{5} + 2)(\sqrt{5} - 2)} $$ $$ x^2 = 2\sqrt{5} + 2\sqrt{(\sqrt{5})^2 - 2^2} $$ $$ x^2 = 2\sqrt{5} + 2\sqrt{5 - 4} = 2\sqrt{5} + 2 $$ * Factor out the $2$: $x^2 = 2(\sqrt{5} + 1)$ * So, the numerator $x = \sqrt{2(\sqrt{5} + 1)} = \sqrt{2} \cdot \sqrt{\sqrt{5} + 1}$ * Now, substitute the simplified numerator back into the fraction: $$ \frac{\sqrt{2} \cdot \sqrt{\sqrt{5} + 1}}{\sqrt{\sqrt{5} + 1}} $$ * The $\sqrt{\sqrt{5} + 1}$ terms cancel out completely, leaving just $\sqrt{2}$. * Simplify the second part of the expression: $\sqrt{3 - 2\sqrt{2}}$ We need numbers adding to $3$ and multiplying to $2$. They are $2$ and $1$. So, $\sqrt{3 - 2\sqrt{2}} = \sqrt{2} - \sqrt{1} = \sqrt{2} - 1$ * Combine both parts to find $N$: $$ N = \sqrt{2} - (\sqrt{2} - 1) $$ $$ N = \sqrt{2} - \sqrt{2} + 1 = 1 $$ Exam Strategy & Shortcut Look for relationships between the numerator and denominator before doing heavy calculations. When you square the numerator, you get $2\sqrt{5} + 2$, which is exactly twice the expression inside the denominator's root ($\sqrt{5} + 1$). Spotting this proportional relationship immediately tells you the whole fraction simplifies to $\sqrt{2}$, saving substantial time. Common Pitfall A common error is mishandling the subtraction of the second term. Writing $\sqrt{2} - \sqrt{2} - 1$ instead of applying the negative sign to the entire group $-(\sqrt{2} - 1)$ will yield $-1$ instead of $1$, which might mistakenly lead to a wrong guess or wasted time recalculating. Final Answer Therefore, the correct answer is 1.