Math, asked by abhisaranya, 11 months ago

. If alpha,betaare
complex cube roots of cunity
then
alpha^100+beta^100+1/alpha^100*beta^100​

Answers

Answered by Anonymous
0

Answer:

        0

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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