Complex numbers, cubic equation, desperate!! "Prove that z5^3-3z5 = z0"

[Helpless1234]

Hello,

The final question of my problem set due in 5 hours eludes me!!

z0 is a complex different from 2'

z^3-3z=z0

z1=(z0+2)/(z0-2)

z2=z1^(1/2) so the square root

z3=(z2+1)/(z2-1)

z4=z3^(1/3) so the cubic root

z5=z4+1/z4

Prove that z5^3-3z5 = z0

I've tried to express z5 in terms of z0 and see if it could be simplified but that didn't really work out. I've been trying to maybe see if something can be done by expressing z exponentially but I'm very stuck.
Thank you for any help!!

 

[Steven G]

Hello,

The final question of my problem set due in 5 hours eludes me!!

z0 is a complex different from 2'

z^3-3z=z0

z1=(z0+2)/(z0-2)

z2=z1^(1/2) so the square root

z3=(z2+1)/(z2-1)

z4=z3^(1/3) so the cubic root

z5=z4+1/z4

Prove that z5^3-3z5 = z0

I've tried to express z5 in terms of z0 and see if it could be simplified but that didn't really work out. I've been trying to maybe see if something can be done by expressing z exponentially but I'm very stuck.
Thank you for any help!!

As our forum rules state (did you read them?) you should show us your work so we can see where you are going wrong. We really can't help you if we don't know where you are stuck.So please post your work.

Hint: You have z5 in terms of z4, z4 in terms of z3, z3 in terms of z2, z2 in terms of z1 and z1 in terms of z0. So you can get z5 in terms of z0. Then verify that z5^3-3z5 = z0