If and are the roots of the equation x^2-x+1=0 , then ²⁰⁰⁹+ ²⁰⁰⁹ is equal to

Step-by-step solution

and are the roots of quadratic equation aligned x^2-x+1 & =0 x^2-x+1 & =0 x & = 1 1-4 2 = 1 -3 2 & = 1 3 i 2 =- and - ^2 aligned [where, w is cube roots of unity] - ,- ^2 are roots of quadratic equation x^2-x+1=0 Let =-w , then =- ^2 Now, ²⁰⁰⁹+ ²⁰⁰⁹ aligned & =(- )²⁰⁰⁹+ (- ^2 )²⁰⁰⁹=- [ ^ 3(669)+2 + ⁴⁰¹⁸ ] & =- [ ( ^3 )⁶⁶⁹ ^2+ ^ 3(1339)+1 ] aligned aligned & =- [ ( ^3 )⁶⁶⁹ ^2+ ( ^3 )¹³³⁹ ] & [ ^3=1 ] & =- [ ^2+ ] & [ 1+ + ^2=0 ] & =-(-1)=1 & aligned