complex cube of unity (2 Viewers)

[*Pooja*]

if w is a complex root of unity, simplify ( 2 + 2w + w^2)^3.

i have forgotten the concept. can someone pls explain how to do this?

 

[KeypadSDM]

I'm assuming from your subject line that w is in fact a cube root of unity.

w³ = 1
(w - 1)(w² + w + 1) = 0
w != 1
:. w² + w + 1 = 0

Ignore the rest if you don't want to hear the answer

:.( 2 + 2w + w²)³
= (2 + 2w + 2w² - w²)³
= (-w²)³
=-w⁶
= -1

 

[CM_Tutor]

Assuming that this means that w is a non-real complex cube root of unity, the w³ = 1, but w <> 1

So, w³ - 1 = 0
(w - 1)(1 + w + w²) = 0

So, 1 + w + w² = 0, as w - 1 <> 0

So, 1 + w = -w² _____ (*)

We want (2 + 2w + w²)³ = [2(1 + w) + w²]³ = [2(-w²) + w²]³, using (*)
= (-w²)³ = (-1)³ * w⁶ = -1 * (w³)² = -1, as w³ = 1

Edit: KeypadSDM and I were clearly typing at the same time

 

[*Pooja*]

thanks to u both!

 

[Teoh]

We want (2 + 2w + w²)³ = [2(1 + w) + w²]³ = [2(-w²) + w²]³, using (*)
= (-w²)³ = (-1)³ * w⁶ = -1 * (w³)² = -1, as w³ = 1

Edit: KeypadSDM and I were clearly typing at the same time [/B]

Might be kinda stoopid, but I didn't understand why we wanted to get here...

 

[Teoh]

Edit: Yes, well that was just a totally stupid post by me