SOLUTION: i90i91+i92i93 note real numbers are exponents that add up to i to the 366 power

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Question 70009: i90i91+i92i93 note real numbers are exponents that add up to i to the 366 power
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Recognize that by definition i%5E2+=+-1 and i%5E1=i
From this we can develop the sequence:
i+=+i
i%5E2+=+-1
i%5E3+=+i%5E2+%2A+i+=+%28-1%29%2Ai+=+-i
i%5E4+=+i%5E2%2Ai%5E2+=+%28-1%29%2A%28-1%29+=+1
i%5E5+=+i%5E4%2Ai+=+1%2Ai+=+i
i%5E6+=+i%5E3%2Ai%5E3+=+%28-i%29%2A%28-i%29+=+i%5E2+=+-1
i%5E7+=+i%5E6%2Ai%5E1+=+%28-1%29%2Ai+=+-i
i%5E8+=+i%5E4%2Ai%5E4+=+1%2A1+=+1
i%5E9+=+i%5E8%2Ai%5E1+=+1%2Ai+=+i
Note that this series is i, -1, -i, 1, i, -1, -i, 1, i, ....
We can use this table along with some rules of exponents to simplify the terms in the problem.
Let's try to simplify i%5E90. We use one of the laws of exponents to rewrite
the this as i%5E90+=+%28i%5E9%29%5E10. From the above table you can see that i%5E9+=+i
Substitute this to get i%5E90+=+%28i%29%5E10. We can further use a law of exponents
to factor this term into i%5E10+=+i%5E5%2Ai%5E5. But i%5E5+=+i so:
i%5E5%2Ai%5E5=i%2Ai=i%5E2+=+-1
So i%5E90=-1.
We can use this result to find i%5E91. Note that i%5E91+=+i%5E90%2Ai%5E1. Substitute
-1 for i%5E90 to get i%5E91+=+%28-1%29%2Ai+=+-i.
Then use this result and procedure to find that:
i%5E92+=+i%5E91%2Ai%5E1+=%28-i%29%2Ai+=+-i%5E2+=+-%28-1%29+=+1
Repeat this process to find that:
i%5E93+=+i%5E92%2Ai%5E1+=+1%2Ai+=+i
Substitute these four results into the original problem:
%28i%5E90%29%2A%28i%5E91%29+%2B+%28i%5E92%29%2A%28i%5E93%29
The substitution results in:
%28-1%29%2A%28-i%29+%2B+%281%29%2A%28i%29+=+i+%2B+i+=+2i
The answer to your problem is 2i.
Hope this helps.