To find the roots of a complex number, we follow the process outlined below, which utilizes polar (or trigonometric) form and De Moivre's Theorem:

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Steps to Find Roots of a Complex Number:

1. Convert the complex number to polar form: A complex number $z = a + bi$ can be written in polar form as: $z = r \text{cis} \theta$ where:

   • $r = \sqrt{a^2 + b^2}$ is the modulus of $z$,
   • $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ is the argument (angle) of $z$,
   • $\text{cis} \theta = \cos \theta + i \sin \theta$.

2. Apply the nth root formula: The $n$-th roots of a complex number are given by: $z_k = r^{1/n} \text{cis} \left(\frac{\theta + 2k\pi}{n}\right), \quad k = 0, 1, 2, \ldots, n-1.$ Here:

   • $r^{1/n}$ is the nth root of the modulus,
   • $\frac{\theta + 2k\pi}{n}$ are the angles for each root,
   • $k$ represents the different roots.

3. Simplify each root: For each $k$, compute $z_k$ using: $z_k = r^{1/n} \left(\cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n}\right).$

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Example:

Find the cube roots of $z = 8$ (purely real).

1. Write $z$ in polar form: $z = 8 \text{cis} 0$ (since $8$ is on the real axis with $\theta = 0$).

2. Apply the nth root formula for $n = 3$: $z_k = 2 \text{cis} \left(\frac{0 + 2k\pi}{3}\right), \quad k = 0, 1, 2.$

3. Compute the roots:

   • For $k = 0$: $z_0 = 2 \text{cis} 0 = 2.$
   • For $k = 1$: $z_1 = 2 \text{cis} \frac{2\pi}{3} = 2 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right).$
   • For $k = 2$: $z_2 = 2 \text{cis} \frac{4\pi}{3} = 2 \left(-\frac{1}{2} - i \frac{\sqrt{3}}{2}\right).$

Thus, the roots are: $z_0 = 2, \quad z_1 = -1 + i\sqrt{3}, \quad z_2 = -1 - i\sqrt{3}.$

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Questions to Deepen Understanding:

1. How is the argument $\theta$ calculated for complex numbers not on the real axis?
2. What changes in the method if the modulus of the complex number is less than 1?
3. Can you explain how De Moivre's Theorem helps generalize this process?
4. How would you find the fourth roots of $1 + i$?
5. What happens if $k$ exceeds $n-1$ in the formula?

Tip: Always sketch the roots on the complex plane to visualize their symmetry. Roots are evenly spaced in a circle centered at the origin.