Find the indicated root. \[ \sqrt[8]{-1} \]

To find the eighth root of \(-1\), we can express \(-1\) in its polar form. In the complex plane, \(-1\) can be written as \(e^{i\pi}\) (since it lies on the negative x-axis). Now, when seeking the eighth roots, we apply De Moivre's Theorem, which states that the \(n\)th roots of a complex number can be found using: \[ z_k = r^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \] For \(-1\), \(r = 1\) and \(\theta = \pi\), and we are looking for the \(n = 8\) roots. Thus, the eighth roots can be calculated as: \[ z_k = 1^{1/8} \left( \cos\left(\frac{\pi + 2k\pi}{8}\right) + i\sin\left(\frac{\pi + 2k\pi}{8}\right) \right) \] where \(k = 0, 1, 2, \ldots, 7\). Calculating for each \(k\): 1. For \(k = 0\): \[\frac{\pi}{8}\] \(z_0 = \cos\left(\frac{\pi}{8}\right) + i\sin\left(\frac{\pi}{8}\right)\) 2. For \(k = 1\): \[\frac{\pi + 2\pi}{8} = \frac{3\pi}{8}\] \(z_1 = \cos\left(\frac{3\pi}{8}\right) + i\sin\left(\frac{3\pi}{8}\right)\) 3. For \(k = 2\): \[\frac{\pi + 4\pi}{8} = \frac{5\pi}{8}\] \(z_2 = \cos\left(\frac{5\pi}{8}\right) + i\sin\left(\frac{5\pi}{8}\right)\) 4. For \(k = 3\): \[\frac{\pi + 6\pi}{8} = \frac{7\pi}{8}\] \(z_3 = \cos\left(\frac{7\pi}{8}\right) + i\sin\left(\frac{7\pi}{8}\right)\) 5. For \(k = 4\): \[\frac{\pi + 8\pi}{8} = \frac{9\pi}{8}\] \(z_4 = \cos\left(\frac{9\pi}{8}\right) + i\sin\left(\frac{9\pi}{8}\right)\) 6. For \(k = 5\): \[\frac{\pi + 10\pi}{8} = \frac{11\pi}{8}\] \(z_5 = \cos\left(\frac{11\pi}{8}\right) + i\sin\left(\frac{11\pi}{8}\right)\) 7. For \(k = 6\): \[\frac{\pi + 12\pi}{8} = \frac{13\pi}{8}\] \(z_6 = \cos\left(\frac{13\pi}{8}\right) + i\sin\left(\frac{13\pi}{8}\right)\) 8. For \(k = 7\): \[\frac{\pi + 14\pi}{8} = \frac{15\pi}{8}\] \(z_7 = \cos\left(\frac{15\pi}{8}\right) + i\sin\left(\frac{15\pi}{8}\right)\) These provide the eight distinct eighth roots of \(-1\). The final results are: 1. \(z_0 = \cos\left(\frac{\pi}{8}\right) + i\sin\left(\frac{\pi}{8}\right)\) 2. \(z_1 = \cos\left(\frac{3\pi}{8}\right) + i\sin\left(\frac{3\pi}{8}\right)\) 3. \(z_2 = \cos\left(\frac{5\pi}{8}\right) + i\sin\left(\frac{5\pi}{8}\