Algebra> If 1,w,w2 are cube roots of unity, prove ...

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

the_game_10 , 10 Years ago

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2 Answers

Y RAJYALAKSHMI

Last Activity: 10 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

Approved

Rishi Sharma

Last Activity: 5 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