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

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Question 194943: please explain how to simplify...

i to the 100th power
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Since i=sqrt%28-1%29, this means that


i%5E1=i


i%5E2=%28sqrt%28-1%29%29%5E2=-1 or i%5E2=-1


i%5E3=%28sqrt%28-1%29%29%5E3=%28sqrt%28-1%29%29%5E2%2Asqrt%28-1%29=-1%2Ai=-i or i%5E3=-i


i%5E4=%28sqrt%28-1%29%29%5E4=%28sqrt%28-1%29%29%5E2%2A%28sqrt%28-1%29%29%5E2=-1%2A-1=1 or i%5E4=1


i%5E5=%28sqrt%28-1%29%29%5E5=%28sqrt%28-1%29%29%5E4%2Asqrt%28-1%29=1%2Ai=i or i%5E5=i


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 i%5Ek (where "k" is a whole number),


If the remainder of k%2F4 is 0, then i%5Ek=1.


If the remainder of k%2F4 is 1, then i%5Ek=i.


If the remainder of k%2F4 is 2, then i%5Ek=-1.


If the remainder of k%2F4 is 3, then i%5Ek=-i.


So out at the 100th term, since 100%2F4=25 remainder 0, this means that i%5E%28100%29=1