1Ian FowlerAug 5, 2020 · Edited: Aug 5, 2020x + 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/2I'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)/2and 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) = -1I'll leave x = sqrt(3)/2 - i/2 for you. You should get the same thing.

1anbowang43Aug 7, 2020@Ian Fowler my question was actually x^1024+x^-2014 not x^1024+x^-1024 but good solution tho

Ian FowlerAug 7, 2020So right you are. Thanks for the correction. Sometimes I'm in too much of a hurry. The same method can be used.

x + 1/x = sqrt(3)

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

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

-1I'll leave

x = sqrt(3)/2 - i/2for you. You should get the same thing.Thx

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

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