Question

if 1,ω,ω^2 are the three cube roots of unity prove that
(1+ω)(1+ω^2)(1+ω^4)(1+ω^5)=9
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Answer 1

Solution :

It is given that 1, ω and ω² are the three cube roots of unity.

So, • 1 + ω + ω² = 0 and ω³ = 1

LHS :

(1+ω)(1+ω^2)(1+ω^4)(1+ω^5)

We know that , 1 + ω = - ω² and 1 + ω² = - ω

Substituting this :

(1+ω)(1+ω^2)(1+ω^4)(1+ω^5)

=> - ω² × - ω × ( 1+ω^4)(1+ω^5)

=> ω³( 1+ω^4)(1+ω^5)

ω³ = 1

=> ( 1+ω^4)(1+ω^5)

=> 1 + ω⁴ + ω⁵ + ω⁹

=> 1 + ω⁴ ( 1 + ω ) + ω⁹

=> 1 - ω⁶ + ω⁹

=> 1 - ω⁶ + 1

=> 2 + ω⁶

=> 2 + 7

=> 9 .

Hence Proved

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Answer 2

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