Let w_0, w_1, ..., w_6 be the seven 7th roots of unity (the roots of…

Question

Let w_0, w_1, ..., w_6 be the seven 7th roots of unity (the roots of z^7 = 1). Compute the sum sum_{k=0}^{6} |2 - w_k|^4. Report the exact integer value.

Accepted Answer

For each root w = e^{i*theta}, |2 - w|^2 = (2 - cos theta)^2 + sin^2 theta = 4 - 4cos theta + 1 = 5 - 4cos theta. Thus |2 - w|^4 = (5 - 4cos theta)^2 = 25 - 40cos theta + 16cos^2 theta. Sum over the seven 7th roots, whose angles theta_k = 2*pi*k/7. Using sum_{k=0}^{6} cos theta_k = 0 (the real part of the sum of all 7th roots is 0) and sum_{k=0}^{6} cos^2 theta_k = 7/2 (since cos^2 = (1 + cos 2theta)/2 and the cos 2theta terms also sum to 0), we get sum |2 - w|^4 = 7*25 - 40*0 + 16*(7/2) = 175 + 56 = 231. (Equivalently, the general formula is n[(a^2 + 1)^2 + 2a^2] with n = 7, a = 2: 7*(25 + 8) = 231.)
Final answer: 231

Final answer: 231

Let w_0, w_1, ..., w_6 be the seven 7th roots of unity (the roots of…

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