Respuesta :

Answer:   -i

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Work Shown:

Method 1

List out the first few powers of 'i', where [tex]i = \sqrt{-1}[/tex]

  • i^0 = 1
  • i^1 = i
  • i^2 = -1
  • i^3 = -i
  • i^4 = 1

The process repeats every four terms. Divide the exponent by 4 to look at the remainder.

39/4 = 9 remainder 3

Ignore the quotient 9. The remainder is all we care about

Since we get remainder 3, this means i^39 = i^3 = -i

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Method 2

i^39 = i^(36+3)

i^39 = i^36*i^3

i^39 = i^(9*4)*i^3

i^39 = (i^4)^9*i^3

i^39 = (1)^9*i^3

i^39 = 1*i^3

i^39 = i^3

i^39 = -i

This is a bit longer method, but it helps confirm that i^39 = i^3 = -i we found earlier in the previous section.

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