If $$x = \sqrt a + \frac{1}{{\sqrt a }}{\text{,}}$$    $$y = \sqrt a - \frac{1}{{\sqrt a }}{\text{,}}$$   $$\left( {a > 0} \right)$$   then the value of x⁴ + y⁴ - 2x²y² is?

Solution (By Examveda Team)

$$\eqalign{ & {\text{ }}x = \sqrt a + \frac{1}{{\sqrt a }} \cr & {\text{ }}y = \sqrt a - \frac{1}{{\sqrt a }} \cr & {\text{Put }}a = 4 \cr & x = 2 + \frac{1}{2} = \frac{5}{2} \cr & y = 2 - \frac{1}{2} = \frac{3}{2} \cr & {\text{Then, }}{x^4} + {y^4} - 2{x^2}{y^2}{\text{ }} \cr & = {\left( {{x^2} - {y^2}} \right)^2} \cr & = {\left( {\frac{{25}}{4} - \frac{9}{4}} \right)^2} \cr & = {\text{ 16}} \cr} $$