If $$N = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 7 + \sqrt 3 }},$$    then what is the value of $$N + \frac{1}{N}?$$

Solution (By Examveda Team)

$$\eqalign{ & N = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 7 + \sqrt 3 }} \cr & = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 7 + \sqrt 3 }} \times \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 7 - \sqrt 3 }} \cr & = \frac{{7 + 3 - 2\sqrt {21} }}{4} \cr & = \frac{{10 - 2\sqrt {21} }}{4} \cr & = \frac{5}{2} - \frac{{\sqrt {21} }}{2} \cr & {\text{Similarly }}\frac{1}{N} = \frac{5}{2} + \frac{{\sqrt {21} }}{2} \cr & \therefore N + \frac{1}{N} \cr & = \frac{5}{2} - \frac{{\sqrt {21} }}{2} + \frac{5}{2} + \frac{{\sqrt {21} }}{2} \cr & = \frac{5}{2} + \frac{5}{2} \cr & = 5 \cr} $$