To find the cube roots of $8i$, let's start by expressing $8i$ in polar form and then using De Moivre's theorem.

1. Convert $8i$ to Polar Form:

   • The modulus $r$ of $8i$ is: $r = |8i| = \sqrt{0^2 + 8^2} = 8$
   • The argument $\theta$ of $8i$ is: $\theta = \tan^{-1}\left(\frac{8}{0}\right) = \frac{\pi}{2}$

   So, in polar form: $8i = 8 \text{cis} \frac{\pi}{2} \quad \text{where cis} \, \theta = \cos \theta + i \sin \theta$

2. Apply De Moivre's Theorem for Cube Roots: To find the cube roots of $8i$, we seek values of $z$ such that: $z^3 = 8 \text{cis} \frac{\pi}{2}$ Since the modulus is $8$, the modulus of each cube root will be: $\sqrt[3]{8} = 2$ and each root will have an argument given by: $\frac{\frac{\pi}{2} + 2\pi k}{3} \quad \text{for} \quad k = 0, 1, 2$

3. Calculate the Three Cube Roots: For each value of $k$, we find a different cube root:

   • For $k = 0$: $z_0 = 2 \text{cis} \frac{\pi}{6} = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) = 2 \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = \sqrt{3} + i$

   • For $k = 1$: $z_1 = 2 \text{cis} \frac{5\pi}{6} = 2 \left( \cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6} \right) = 2 \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = -\sqrt{3} + i$

   • For $k = 2$: $z_2 = 2 \text{cis} \frac{3\pi}{2} = 2 \left( \cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2} \right) = 2(0 - i) = -2i$

4. Final Answer: The three cube roots of $8i$ are: $z_0 = \sqrt{3} + i, \quad z_1 = -\sqrt{3} + i, \quad z_2 = -2i$

Would you like further details or explanations on any part of this?

Related Questions

1. How can we generalize the process for finding nth roots of a complex number?
2. What is the geometric representation of these cube roots on the complex plane?
3. How does De Moivre's theorem work for negative arguments or negative moduli?
4. How do we express any complex number purely in polar form?
5. Can we verify these cube roots by direct multiplication?

Tip: When finding cube roots, using polar form and De Moivre’s theorem simplifies the calculation, especially for complex numbers with non-zero imaginary parts.