If 1,ω and ω^2 are the cube roots of unity, prove that (1−ω)(1−ω^2 )(1−ω^4 )(1−ω^5 )=9


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Question Number 136346 by ZiYangLee last updated on 21/Mar/21

$If 1, ω and ω^{2} are the cube roots of unity,$ $prove that (1 - ω) (1 - ω^{2}) (1 - ω^{4}) (1 - ω^{5}) = 9$
	Answered by Rasheed.Sindhi last updated on 21/Mar/21
	$(1−ω)(1−ω^2 )(1−ω^4 )(1−ω^5 ) (1−ω)(1−ω^2 )(1−ω^3 .ω)(1−ω^3 .ω^2 ) ={(1−ω)(1−ω^2 )}{(1−ω)(1−ω^2 )} =(1−ω−ω^2 +ω^3 )^2 =(1−(ω+ω^2 )+ω^3 )^2 =(1−(−1)+1)^2 =3^2 =9 Proved$
	$(1 - ω) (1 - ω^{2}) (1 - ω^{4}) (1 - ω^{5})$ $(1 - ω) (1 - ω^{2}) (1 - ω^{3} . ω) (1 - ω^{3} . ω^{2})$ $= {(1 - ω) (1 - ω^{2})} {(1 - ω) (1 - ω^{2})}$ $= {(1 - ω - ω^{2} + ω^{3})}^{2}$ $= {(1 - (ω + ω^{2}) + ω^{3})}^{2}$ $= {(1 - (- 1) + 1)}^{2} = 3^{2} = 9$ $P r o v e d$
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