Evaluate [i¹⁸+ ( 1 i )²⁵ ]³ .

Step-by-step solution

i¹⁸ = i¹⁶ i^2 = 1 (-1) = -1 1 i = i i^2 = i -1 = -i ( 1 i )²⁵ = (-i)²⁵ = (-1)²⁵ i²⁵ = -1 i²⁴ i = -i [i¹⁸ + ( 1 i )²⁵ ]^3 = [-1 + (-i)]^3 = [-(1+i)]^3 = -(1+i)^3 (1+i)^2 = 1 + 2i + i^2 = 2i (1+i)^3 = 2i(1+i) = 2i + 2i^2 = 2i - 2 = -2 + 2i -(1+i)^3 = -(-2+2i) = 2 - 2i = 2(1-i) Correct option: (1)