The smallest positive integer n for which (1 + i)^n (1 - i)^ n-2 is a real number, is ...

Step-by-step solution

The given expression is (1 + i)^n (1 - i)^ n-2 . This can be rewritten as ( 1 + i 1 - i )^n (1 - i)^2 . Simplifying the term inside the bracket: 1 + i 1 - i = (1 + i)^2 (1 - i)(1 + i) = 1 + i^2 + 2i 1^2 - i^2 = 2i 2 = i Simplifying the second term: (1 - i)^2 = 1 + i^2 - 2i = -2i Substituting these back into the expression, we get: i^n (-2i) = -2 i^ n+1 For this expression to be a real number, i^ n+1 must be a real number. This is possible only when the exponent n+1 is an even integer. n + 1 = 2k , where k is an int