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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
$$\mathrm{If}\:\mathrm{1},\omega\:\mathrm{and}\:\omega^{\mathrm{2}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}, \\ $$$$\mathrm{prove}\:\mathrm{that}\:\left(\mathrm{1}−\omega\right)\left(\mathrm{1}−\omega^{\mathrm{2}} \right)\left(\mathrm{1}−\omega^{\mathrm{4}} \right)\left(\mathrm{1}−\omega^{\mathrm{5}} \right)=\mathrm{9} \\ $$
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
$$\left(\mathrm{1}−\omega\right)\left(\mathrm{1}−\omega^{\mathrm{2}} \right)\left(\mathrm{1}−\omega^{\mathrm{4}} \right)\left(\mathrm{1}−\omega^{\mathrm{5}} \right) \\ $$$$\left(\mathrm{1}−\omega\right)\left(\mathrm{1}−\omega^{\mathrm{2}} \right)\left(\mathrm{1}−\omega^{\mathrm{3}} .\omega\right)\left(\mathrm{1}−\omega^{\mathrm{3}} .\omega^{\mathrm{2}} \right) \\ $$$$=\left\{\left(\mathrm{1}−\omega\right)\left(\mathrm{1}−\omega^{\mathrm{2}} \right)\right\}\left\{\left(\mathrm{1}−\omega\right)\left(\mathrm{1}−\omega^{\mathrm{2}} \right)\right\} \\ $$$$=\left(\mathrm{1}−\omega−\omega^{\mathrm{2}} +\omega^{\mathrm{3}} \right)^{\mathrm{2}} \\ $$$$=\left(\mathrm{1}−\left(\omega+\omega^{\mathrm{2}} \right)+\omega^{\mathrm{3}} \right)^{\mathrm{2}} \\ $$$$=\left(\mathrm{1}−\left(−\mathrm{1}\right)+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{3}^{\mathrm{2}} =\mathrm{9} \\ $$$${Proved} \\ $$

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