If = -1 + -3 2 , then what is the value of (1 + ¹⁹ - ³⁵)¹⁰⁰ - (1 - 3 ²⁵ + ³⁸)⁵⁰ ?

Step-by-step solution

Given = -1 + -3 2 , which is the complex cube root of unity, . We know that ^3 = 1 and 1 + + ^2 = 0 . Substituting = into the given expression, we simplify the powers of : ¹⁹ = ¹⁹ = ( ^3)^6 = ³⁵ = ³⁵ = ( ^3)¹¹ ^2 = ^2 ²⁵ = ²⁵ = ( ^3)^8 = ³⁸ = ³⁸ = ( ^3)¹² ^2 = ^2 The first term becomes: (1 + ¹⁹ - ³⁵)¹⁰⁰ = (1 + - ^2)¹⁰⁰ Since 1 + = - ^2 , this reduces to: (- ^2 - ^2)¹⁰⁰ = (-2 ^2)¹⁰⁰ = 2¹⁰⁰ ²⁰⁰ Since 200 = 3 66 + 2 , ²⁰⁰ = ^2 . Thus, the first term is 2¹⁰⁰ ^2 . The second term becomes: (1 - 3 ²⁵ + ³⁸)⁵⁰ = (1 - 3 + ^2