Let z be a complex number such that z^2 + z + 1 = 0 . The value of _ r=1 ⁸ ⁸C_ r ( z^r + 1 z^r )^2 is equal to _____.

Step-by-step solution

Given z^2 + z + 1 = 0 , the roots are the complex cube roots of unity. Let z = . The expression inside the summation is: ( z^r + 1 z^r )^2 = z^ 2r + 1 z^ 2r + 2 = ^ 2r + ^ -2r + 2 Substituting this into the summation: _ r=1 ⁸ ⁸C_ r ( ^ 2r + ^ -2r + 2 ) = _ r=1 ⁸ ⁸C_ r ( ^2)^r + _ r=1 ⁸ ⁸C_ r ( ⁻²)^r + 2 _ r=1 ⁸ ⁸C_ r Using the binomial expansion _ r=1 ^ n ^ n C_ r x^r = (1+x)^n - 1 , we get: _ r=1 ⁸ ⁸C_ r ( ^2)^r = (1+ ^2)^8 - 1 _ r=1 ⁸ ⁸C_ r ( ⁻²)^r = (1+ ⁻²)^8 - 1 2 _ r=1 ⁸ ⁸C_ r = 2(2^8 - 1) = 2(256 - 1) = 510 W