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Powers of i - Problem 1 2,881 views

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Simplifying powers of i. So when we’re simplifying powers of i what we need to do is figure out what multiples of 4 are close to this, because we know that i to the 4th and any multiple of 4 there after are going to be the same thing as 1. There’s two ways of doing that. When we’re dealing with a simple number that we know we can just figure out what the closest multiple of 4 is. So in the case of 120 I know that 4 times 30, sorry, in the case of 121, I went a little bit a head of myself, I know that 4 times 30 is 120. Therefore I know that i to the 120 is just equal to 1 because it’s a multiple of 4. I have more i after that so this is just i to the 121 is just going to be 1 times i or i.

The other way to do this is we'll pretend that I didn’t know that 4 times 30 was 120 so therefore we’re close to 121. What you can do is just look at 121 divided by 4. Divide this out long division, 4 goes into 12 three times and we’re left with zero 1 and so doesn’t go into it at all, subtract zero leaves this with a remainder of 1. That remainder is the power of i we’re still concerned with. So this tells me, this remainder of 1 tells me i to the 121 is the same thing as i to the first which I know to be just i.

If I had a remainder of 3 I would know that we had, remainder of 3 would imply that we’re dealing with i to the third which I can remember is –i.

So two different ways if you have a number that is close to a number that you know is a multiple of 4, you can just take the difference, if you don’t know it, just divide it up and the remainder is the power of i you’re concerned with.