Evaluate [i²²+ ( 1 i )²⁵ ]^3

Step-by-step solution

Given the expression [i²² + ( 1 i )²⁵ ]^3 . First, simplify the powers of i . We know that i^2 = -1 , i^4 = 1 , and 1 i = -i . i²² = (i^4)^5 i^2 = 1^5 (-1) = -1 . ( 1 i )²⁵ = (-i)²⁵ = (-1)²⁵ i²⁵ = -1 (i^4)^6 i = -1 1^6 i = -i . Substituting these values into the expression: [-1 + (-i) ]^3 = [-(1 + i) ]^3 = -1 (1 + i)^3 . Expanding (1 + i)^3 using the formula (a + b)^3 = a^3 + b^3 + 3ab(a + b) : (1 + i)^3 = 1^3 + i^3 + 3(1)(i)(1 + i) = 1 - i + 3i(1 + i) = 1 - i + 3i + 3i^2 = 1 + 2i - 3 = -2 + 2i . Finally, multiplyi