<s>[INST] Compute all roots of $(-8)^{\frac{1}{3}}$ $$(-8)^{\frac{1}{3}}$$
The problem states to compute all roots of the complex number above. Below is my attempt, but my inquiries are if I did it right and why it doesn't match Wolfram. Wolfram only returns $1$ result - was I only supposed to have $1$ as well??
$$\text{modulus} = 8$$
$$\theta = \arctan \left(\frac{0}{-8}\right) = 0$$
Because we are working in the left half of the complex plane, $\theta=\pi$.
$$z=\left(8e^{i(2\pi k+\pi)}\right)^{\frac{1}{3}}$$
This complex number has $3$ roots, so $k=0,1,2$ :
$$z_0=2e^{i\frac{\pi}{3}}$$
$$z_1=2e^{i\pi}$$
$$z_2=2e^{i\frac{5\pi}{3}}$$
 [/INST] We can use elementary factorization to find all roots. Let $x = \left(-8\right)^{\frac{1}{3}}\Rightarrow x^3 = -8\Rightarrow x^3 + 8 = 0\Rightarrow (x+2)(x^2-2x+4) = 0\Rightarrow (x+2)((x-1)^2 + 3) = 0\Rightarrow x = -2, (x-1)^2 = -3\Rightarrow x = -2, x-1 = \pm i\sqrt{3}\Rightarrow x = -2, 1 \pm i\sqrt{3}$.
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