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Question

If x=a+b,y=aα+bβandz=aβ+, where α,β not equal 1are cube roots of unity, then xyz equals


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Solution

Find the value of xyz:

Given,

x=a+b,y=aα+bβandz=aβ+, and α,β are cube roots of unity.

Then,

α=ω,β=ω2

Now,

xyz=(a+b)(aα+bβ)(aβ+bα)=(a+b)a2αβ+abα2+abβ2+b2αβ=(a+b)a2αβ+ab(α2+β2)+b2αβ

Now substitute values of α,β

So,

xyz=(a+b)(a2(ωω2)+ab(ω2+ω4)+b2(ωω2))=(a+b)(a2-ab+b2)[ω3=1]=a3+b3

Hence, the correct answer is a3+b3.


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