SOLUTION: i^55

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Question 133841: i^55
Answer by solver91311(12126) About Me  (Show Source):
You can put this solution on YOUR website!
red%28i%5E0%29=1 because anything to the zero power is 1
red%28i%5E1%29=i because anything to the 1 power is itself
red%28i%5E2%29=-1 by definition of i
red%28i%5E3%29=i%5E2%2Ai%5E1=-1%2Ai=-i because a%5En%2Aa%5Em=a%5E%28n%2Bm%29
i%5E4=i%5E2%2Ai%5E2=-1%2A-1=1=red%28i%5E0%29
i%5E5=i%5E4%2Ai%5E1=1%2Ai=i=red%28i%5E1%29
i%5E6=i%5E4%2Ai%5E2=1%2A-1=-1=red%28i%5E2%29, and
i%5E7=i%5E4%2Ai%5E3=1%2A%28-i%29=%28-i%29=red%28i%5E3%29

And the pattern repeats forever.

So, the process is to take the exponent on i, 55 in this case and perform integer division with a divisor of 4. Integer division is the first kind of division you learned in the 2nd or 3rd grade where you got a quotient and a remainder. Then raise i to the power of the remainder.

55%2F4=13r3, so i%5E55=i%5E3=-i