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Simplifying Powers of i

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let's practice simplifying powers of I we learned in class that the powers of I from first power to 4th power RI to the 1st equals II Square was -1 I to the third equals negative R and I to the fourth equals one and I have those listed on this side of the page so then we discovered that we can simplify all of the other powers of I based on this pattern for example I to V could be Rewritten as I to the fourth times I now why do I want to use I to the fourth out of the 4th is my favorite because I to the fourth equals one so for every I to the fourth that I can use I can replace it with just a plain old one so I'm going to substitute in the place of I to the fourth one and so now I just have one eye to the first which is going to equal whatever I to the first equals which is just I so let's look at one with a bigger power let's try I to the 17th so for I to the 17th I need a whole bunch more fours so I to the four times I to the fourth that gives me a total of 8 another I to the fourth gives me a total of 12 henna personal items for give me a total of 16 I can't use another I to the 4th because that would put me over 17 so right now I got a total of 16 I just need one more so I to the fur is all that's left so each of these I to the fourth power here here here here those are all just one's 1 times 1 times 1 times 1 equals what so that gives me one time I took the first which again is just I so what is the number is really big and I don't want to ride out all is i to the fourth will notice that's what's happening is I'm just going through 4 + 4 + 4 + 4 + all that really matters is what's left over after I divided by 4 so for this example with 17 I could have just divided 17 by 4 4 goes into 17 four times with a remainder of 1 so it ends up that I to the first is how the expression will simplify so when I get to a bigger number let's do I to the 23rd instead of writing out all those powers of for I can just divide 54023 / 4 for can go fully in 22035 * 5 * 4 is 20 so all of these would be I would have five powers of 4 and then my remainder is 3 all I really care about the remainder is the remainder because all the powers of four equal one so I took the 3rd is going to be my answer and I to the Third negative I not another thing that I have seen students do is to basically discount so I could if I was trying to do I to the 23rd I could go two three four five six seven eight nine 10 11 12 13 14 15 16 17 18 19 20 21 22 23 but it seems faster me to divide but you can always fall back on Counting you have the powers of I memorized in order

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Simplifying Powers of i

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