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COMEDK2025MathematicsComplex NumberActual

The value of (- -1 )^ 4 n-3 , where n N , set of natural numbers is

Options

  1. A1
  2. B-1
  3. C-i
  4. Di

Correct answer

C. -i

Step-by-step solution

Given the expression (- -1 )^ 4n-3 , where n N . Since -1 = i , the expression becomes (-i)^ 4n-3 . This can be written as (-1)^ 4n-3 i^ 4n-3 . Since 4n-3 is an odd number for any n N , (-1)^ 4n-3 = -1 . Now, i^ 4n-3 = i^ 4n i⁻³ = (i^4)^n 1 i^3 . Since i^4 = 1 and i^3 = -i , we have i^ 4n-3 = 1^n 1 -i = 1 -i = i -i^2 = i 1 = i . Therefore, (-i)^ 4n-3 = -1 i = -i . Answer: -i

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