SOLUTION: can you please explain the answer to i^49? I factor out 13*3, but cannot resolve the remaining terms...thanks

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Question 172894: can you please explain the answer to i^49? I factor out 13*3, but cannot resolve the remaining terms...thanks
Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(21611) About Me  (Show Source):
You can put this solution on YOUR website!
can you please explain the answer to i^49? I factor out 13*3, but cannot resolve the remaining terms...thanks
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That's not clear. 13*3 = 39. What does that have to do with anything?
Is i = sqrt(-1)? Or is it just a variable?

Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
Take note that 49 = 48 + 1 and 48 = 12*4.

Also, we'll use the identities x%5E%28y%2Bz%29=x%5Ey%2Ax%5Ez and i%5E4=1


So i%5E%2849%29=i%5E%2848%2B1%29=i%5E%2812%2A4%2B1%29=i%5E%2812%2A4%29i%5E1=%28i%5E4%29%5E12%2Ai=%281%29%5E12%2Ai=%281%29%2Ai=i


This means that i%5E49=i

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If you want an easier method, do the following:


Divide the exponent 49 by 4 to get 12 remainder 1


Since the remainder is 1, this tells us that the answer is "i" (ie i%5E49=i)



This is a handy thing to remember (or put in your reference book)

If the remainder is 0, the answer is 1
If the remainder is 1, the answer is "i"
If the remainder is 2, the answer is -1
If the remainder is 3, the answer is "-i"