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.