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If-1-and-2-are-the-cube-roots-of-unity-prove-that-1-1-2-1-4-1-5-9-

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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
If1,ωandω2arethecuberootsofunity,provethat(1−ω)(1−ω2)(1−ω4)(1−ω5)=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−ω2)(1−ω4)(1−ω5)(1−ω)(1−ω2)(1−ω3.ω)(1−ω3.ω2)={(1−ω)(1−ω2)}{(1−ω)(1−ω2)}=(1−ω−ω2+ω3)2=(1−(ω+ω2)+ω3)2=(1−(−1)+1)2=32=9Proved

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