Given equation $\frac{z^7 + 1}{z^3 + 1} = 0$ where $z \neq -1$. Find $z$ and the sum of all values.
Step-by-Step Solution
Key Concept: Use de Moivre's theorem to find complex roots and identify which roots satisfy additional constraints
From $z^7 + 1 = 0$, we have $z^7 = -1$, so $z = e^{i\pi(2k+1)/7}$ for $k = 0, 1, 2, \ldots, 6$. Since $z^3 + 1 \neq 0$, we exclude $z^3 = -1$, which occurs when $k = 5, 7, 10, 12, 13$. The valid values of $k$ are $0, 1, 2, 3, 4, 6$, giving 6 solutions. The angles are $\frac{\pi}{7}, \frac{3\pi}{7}, \frac{5\pi}{7}, \frac{7\pi}{7}, \frac{9\pi}{7}, \frac{11\pi}{7}, \frac{13\pi}{7}$, and the sum of all values equals $16$.
Correct Answer: 16