# A Level Further Maths Powers of i- complex numbers

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how do you work out powers of i say for example I^251 without a calculator like this homework I have got is asking

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how do you work out powers of i say for example I^251 without a calculator like this homework I have got is asking

**liamlarner**)how do you work out powers of i say for example I^251 without a calculator like this homework I have got is asking

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**liamlarner**)

how do you work out powers of i say for example I^251 without a calculator like this homework I have got is asking

So you might want to rewrite 251 in the form 4k + r where k is an integer and r<4 is the remainder.

Then we have: i^251 = i^(4k+r) = (i^4)^k i^r = i^r

TLDR; i to any power is equivalent to i raised to the remainder when dividing the power by 4.

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Every 4th power of i is 1 since i^4 = 1.

So you might want to rewrite 251 in the form 4k + r where k is an integer and r<4 is the remainder.

Then we have: i^251 = i^(4k+r) = (i^4)^k i^r = i^r

TLDR; i to any power is equivalent to i raised to the remainder when dividing the power by 4.

**RDKGames**)Every 4th power of i is 1 since i^4 = 1.

So you might want to rewrite 251 in the form 4k + r where k is an integer and r<4 is the remainder.

Then we have: i^251 = i^(4k+r) = (i^4)^k i^r = i^r

TLDR; i to any power is equivalent to i raised to the remainder when dividing the power by 4.

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this is hella confusing me is their an easier way to do it maybe binomial expansion

**liamlarner**)this is hella confusing me is their an easier way to do it maybe binomial expansion

Example: i^18 … divide 18 by 4 to get remainder 2 … so i^18 = i^2 = -1

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Seriously?

Example: i^18 … divide 18 by 4 to get remainder 2 … so i^18 = i^2 = -1

**RDKGames**)Seriously?

Example: i^18 … divide 18 by 4 to get remainder 2 … so i^18 = i^2 = -1

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#7

i^2=-1, i^250=-1^125, negative power so stays as -1 as with a positive power it’d be 1. Then multiply -1 by i^1 cos you got 1 left and you getttttttttt -i

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