If and are the roots of the equation x^2 + x + 1 = 0 then ²⁰²⁶ + ²⁰²⁶ =

Step-by-step solution

The roots of the equation x^2 + x + 1 = 0 are the complex cube roots of unity, and ^2 . Let = and = ^2 . Substituting the values of and into the given expression gives ²⁰²⁶ + ( ^2)²⁰²⁶ = ²⁰²⁶ + ⁴⁰⁵² . Since ^3 = 1 , dividing the exponents by 3 yields the remainders. 2026 = 3 675 + 1 ²⁰²⁶ = 4052 = 3 1350 + 2 ⁴⁰⁵² = ^2 Therefore, ²⁰²⁶ + ²⁰²⁶ = + ^2 . Using the property 1 + + ^2 = 0 , we get + ^2 = -1 . Thus, ²⁰²⁶ + ²⁰²⁶ = -1 .