Question

If N = [{√(√5+2) + √(√5-2)}/√(√5+1)] - √(√5+1), then find the value of N.

Answer 1

Rationalisation

Given, $N=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}-\sqrt{\sqrt{5}-1}}$

We rationalize the denominator of the right hand side of the given term by multiplying its conjugate irrational number $\sqrt{\sqrt{5}+1}+\sqrt{\sqrt{5}-1}$

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

Now we calculate the denominator using algebraic identity $a^{2}-b^{2}=(a+b)(a-b)$

$\quad N=\frac{\sqrt{(\sqrt{5}+2)(\sqrt{5}+1)}+\sqrt{(\sqrt{5}+2)(\sqrt{5}-1)}+\sqrt{(\sqrt{5}-2)(\sqrt{5}+1)}+\sqrt{(\sqrt{5}-2)(\sqrt{5}-1)}}{\sqrt{5}+1-\sqrt{5}+1}$

$\Rightarrow N=\frac{1}{2}\{\sqrt{7+3\sqrt{5}}+\sqrt{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}+\sqrt{3-3\sqrt{5}}\}$

This is the required value of $N$