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I have tried to solve the problem in the following way:

$$i^{-39} = 1/i^{39} = i^{4\cdot 9+3} = 1/(-i) = -i$$

However the answer is supposed to be $i$

Did I do something wrong?

Calvin Khor
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MsBonniePython
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    please type in mathjax. Also, $1/(-i)=i$ (further, an intermediate step was typed incorrectly but I did not fix it) – Calvin Khor Sep 24 '21 at 05:25

2 Answers2

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Your answer is as follows: $$i^{-39} = \frac{1}{i^{39}} = i^{4\cdot9 + 3} = \frac{1}{-i} = -i.$$

I think the second and last equations should be $$\frac{1}{i^{39}} = \frac{1}{i^{4\cdot9 + 3}} = \frac{1}{-i} = i,$$ since $-i^2 = -(-1) = 1.$

Jose Arnaldo Bebita Dris
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Your very last step was incorrect. $$ \frac{1}{-i}\neq -i $$ Instead, try "multiplying by 1"... $$ \frac{1}{-i}\cdot \Big(1\Big)=\frac{1}{-i}\cdot\Big(\frac{i}{i}\Big)=\frac{i}{-i^2}=\frac{i}{1}=i $$

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