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If 1,w,w2 are cube roots of unity, prove that (1-w)(1-w2)(1-w4)(1-w5) = 9

If 1,w,w2 are cube roots of unity, prove that
(1-w)(1-w2)(1-w4)(1-w5) = 9

Grade:

2 Answers

Y RAJYALAKSHMI
45 Points
9 years ago
We have 1 + w + w2 = 0 & w3 = 1
(1 – w)(1 – w2)(1 – w4)(1 – w5) = (1 – w)(1 – w2)(1 – w3.w)(1 – w3.w2)
= (1 – w)(1 – w2)(1 – w)(1 – w2)  (Since w3 = 1)
= (1 – w)2(1 – w2)2
=(1 – 2w + w2)(1 – 2w2 + w4) = (1 – 2w + w2) (1 – 2w2 – w3.w) = (1 – 2w + w2) (1 – 2w2 – w) 
= ( – w – 2w)( – w2 – 2w2)  (since 1 + w + w2 = 0)
= ( – 3w) ( – 3w2) = 9w3 = 9
 
 
Rishi Sharma
askIITians Faculty 646 Points
3 years ago
Dear Student,
Please find below the solution to your problem.

We have 1 + w + w2 = 0 & w3 = 1
(1 – w)(1 – w2)(1 – w4)(1 – w5) = (1 – w)(1 – w2)(1 – w3.w)(1 – w3.w2)
= (1 – w)(1 – w2)(1 – w)(1 – w2) (Since w3 = 1)
= (1 – w)2(1 – w2)2
=(1 – 2w + w2)(1 – 2w2 + w4) = (1 – 2w + w2) (1 – 2w2 – w3.w) =(1 – 2w + w2) (1 – 2w2 – w)
= ( – w – 2w)( – w2 – 2w2) (since1 + w + w2= 0)
= ( – 3w) ( – 3w2) = 9w3 = 9

Thanks and Regards

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