SOLUTION: please explain how to simplify... i to the 100th power

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Since , this means that





or


or


or


or


etc...

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So for every increment of the exponent, the powers of "i" are: i, -1, -i, 1, i, etc... and this pattern repeats every 4 terms (notice how the 5th term is identical to the first term).


This pattern can be generalized to the following:

For (where "k" is a whole number),


If the remainder of is 0, then .


If the remainder of is 1, then .


If the remainder of is 2, then .


If the remainder of is 3, then .


So out at the 100th term, since remainder 0, this means that