If $$x = a + \frac{1}{a}$$   and $$y = a - \frac{1}{a},$$   then the value of x⁴ + y⁴ - 2x²y² is?

Solution (By Examveda Team)

$$\eqalign{ & x = a + \frac{1}{a} \cr & y = a - \frac{1}{a} \cr & \therefore \left( {x + y} \right) = a + \frac{1}{a} + a - \frac{1}{a} = 2a \cr & \therefore \left( {x - y} \right) = a + \frac{1}{a} - a + \frac{1}{a} = \frac{2}{a} \cr & \therefore {x^4}{\text{ + }}{{\text{y}}^4} - 2{x^2}{y^2} \cr & = {\left( {{x^2} - {y^2}} \right)^2} \cr & = {\left[ {\left( {x + y} \right)\left( {x - y} \right)} \right]^2} \cr & = {\left( {2a \times \frac{2}{a}} \right)^2} \cr & = {\left( 4 \right)^2} \cr & = 16 \cr} $$