Assuming that this means that w is a non-real complex cube root of unity, the w<sup>3</sup> = 1, but w <> 1
So, w<sup>3</sup> - 1 = 0
(w - 1)(1 + w + w<sup>2</sup>) = 0
So, 1 + w + w<sup>2</sup> = 0, as w - 1 <> 0
So, 1 + w = -w<sup>2</sup> _____ (*)
We want (2 + 2w + w<sup>2</sup>)<sup>3</sup> = [2(1 + w) + w<sup>2</sup>]<sup>3</sup> = [2(-w<sup>2</sup>) + w<sup>2</sup>]<sup>3</sup>, using (*)
= (-w<sup>2</sup>)<sup>3</sup> = (-1)<sup>3</sup> * w<sup>6</sup> = -1 * (w<sup>3</sup>)<sup>2</sup> = -1, as w<sup>3</sup> = 1
Edit: KeypadSDM and I were clearly typing at the same time
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