. If alpha,betaare
complex cube roots of cunity
then
alpha^100+beta^100+1/alpha^100*beta^100
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Step-by-step explanation:
As α is a cube root of unity, this means α³ = 1. So...
- α¹⁰⁰ = ( α³ )³³ × α = α
Similarly
- β¹⁰⁰ = ( β³ )³³ × β = β
So the given expression reduces to
- ( α + β + 1 ) / ( αβ ) ... (*)
As they are the complex cube roots of unity, α and β are the roots of the quadratic x² + x + 1. This follows from the fact that x³-1=(x-1)(x²+x+1).
From the coefficients of this quadratic, the sum of the roots is -1 and the product is 1. Thus
- α + β = -1 and αβ = 1
Putting these into the expression (*) gives the answer
( - 1 + 1 ) / 1 = 0 / 1 = 0
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