Answer: -i
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Work Shown:
Method 1
List out the first few powers of 'i', where
- 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.