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anbowang32
Aug 05, 2020

Algebra equation

in Math Problems


3 answers1 reply
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Ian Fowler
Aug 05, 2020  ·  Edited: Aug 05, 2020

x + 1/x = sqrt(3)

Multiply by x and use the Quadratic Formula. x is not real.


x = sqrt(3)/2 + i/2 or x = sqrt(3) - i/2

I'll use the first one.

in polar form x = sqrt(3) /2 + i/2 = 1*cis(pi/6)

Therefore by DeMoivre: x^1024 = 1*cis(1024pi/6)

= cis(512pi/3)

= cis(2pi/3)

= -1/2 + isqrt(3)/2


and x^-1024

= 1*cis(-1024pi/6)

= cis(-512pi/3)

= cis(-2pi/3) =

-1/2 - i sqrt(3)


And we get x^1024 + x^-1024 = (-1/2 +isqrt(3)/2) + (-1/2 - isqrt(3)/2) = -1


I'll leave x = sqrt(3)/2 - i/2 for you. You should get the same thing.





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anbowang32
Aug 06, 2020

Thx

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anbowang43
Aug 07, 2020

@Ian Fowler my question was actually x^1024+x^-2014 not x^1024+x^-1024 but good solution tho

Ian Fowler
Aug 07, 2020

So right you are. Thanks for the correction. Sometimes I'm in too much of a hurry. The same method can be used.

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