Question

if 1, w w² is cube root of 1 then (1-w)(1-w²)(1-w⁴)(1-w⁸)=​

Answer 1

Given : 1, w w² is cube root of 1

To Find : (1-w)(1-w²)(1-w⁴)(1-w⁸)

Solution:

1, w w² is cube root of 1

=> x³ = 1

=> x³ - 1  = 0

1 + w + w² = -0/1 = 0

1.w.w² = w³ = -(-1)/1 = 1

1 + w + w² =  0  =>   w + w² = - 1

w³  = 1

(1-w)(1-w²)(1-w⁴)(1-w⁸)

w⁴ = w³.w = w

w⁸ = w³.w³.w² = w²

= (1-w)(1-w²)(1-w)(1-w²)

= ((1 - w)(1 - w²))²

= (1 - w - w² + w³)²

= ( 1 -(w + w²) + w³)²

w + w² = - 1  , w³= 1

= ( 1 - (-1) + 1)²

= 3²

= 9

(1-w)(1-w²)(1-w⁴)(1-w⁸)  = 9

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