A Level Further Maths Powers of i- complex numbers

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liamlarner
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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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Eris13696
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(Original post by 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
Break it down, so if I had i^6 for example, using the rule of indices, I would do i^6= (i)^3 x(i)^3, I think for bigger powers it would be much exhausting, but good luck
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RDKGames
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(Original post by 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
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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liamlarner
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(Original post by 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.
this is hella confusing me is their an easier way to do it maybe binomial expansion
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RDKGames
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(Original post by liamlarner)
this is hella confusing me is their an easier way to do it maybe binomial expansion
Seriously?

Example: i^18 … divide 18 by 4 to get remainder 2 … so i^18 = i^2 = -1
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liamlarner
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(Original post by RDKGames)
Seriously?

Example: i^18 … divide 18 by 4 to get remainder 2 … so i^18 = i^2 = -1
ah that example helps so if you have i^253 then, 253 = (4x63)+1 so i^253=i
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RandomE
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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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