This problem involves solving the equation $z^4 = 16i$ where $z$ is a complex number. To solve for the four roots of the equation and express them in polar form, we will follow these steps:

Part (a) Find $z_1, z_2, z_3,$ and $z_4$

We are given that the equation has four roots of the form $z_i = r(\cos \theta_i + i \sin \theta_i)$, where $r > 0$ and the angles $\theta_1, \theta_2, \theta_3, \theta_4$ are spaced evenly between 0 and $2\pi$. The right-hand side of the equation is $16i$, which we will express in polar form as well.

1. Express $16i$ in polar form:

   The complex number $16i$ can be written in polar form as:

   $16i = 16 \cdot (\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})$ So, $r = 16$ and $\theta = \frac{\pi}{2}$.

2. Find the four roots using De Moivre's Theorem:

   For the equation $z^4 = 16i$, we take the fourth roots of $16i$, which involves dividing the angle $\frac{\pi}{2}$ by 4 and then adding multiples of $\frac{2\pi}{4}$ to get the four angles.

   • $\theta_1 = \frac{\frac{\pi}{2} + 2k\pi}{4}$ for $k = 0, 1, 2, 3$

   Now, compute these angles:

   • $\theta_1 = \frac{\pi}{8}$
   • $\theta_2 = \frac{\pi}{8} + \frac{\pi}{2} = \frac{5\pi}{8}$
   • $\theta_3 = \frac{\pi}{8} + \pi = \frac{9\pi}{8}$
   • $\theta_4 = \frac{\pi}{8} + \frac{3\pi}{2} = \frac{13\pi}{8}$

3. Find the magnitudes:

   Since $r^4 = 16$, we find $r = \sqrt[4]{16} = 2$.

   So the roots are:

   • $z_1 = 2 \left( \cos \frac{\pi}{8} + i \sin \frac{\pi}{8} \right)$
   • $z_2 = 2 \left( \cos \frac{5\pi}{8} + i \sin \frac{5\pi}{8} \right)$
   • $z_3 = 2 \left( \cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8} \right)$
   • $z_4 = 2 \left( \cos \frac{13\pi}{8} + i \sin \frac{13\pi}{8} \right)$

Part (b) Find the common ratio of the geometric sequence

Since the four roots $z_1, z_2, z_3, z_4$ form a geometric sequence, the common ratio can be found by dividing any root by its preceding root. For instance, the common ratio $r$ is:

$r = \frac{z_2}{z_1} = \frac{2 \left( \cos \frac{5\pi}{8} + i \sin \frac{5\pi}{8} \right)}{2 \left( \cos \frac{\pi}{8} + i \sin \frac{\pi}{8} \right)}$

This can be simplified to:

$r = \frac{\cos \frac{5\pi}{8} + i \sin \frac{5\pi}{8}}{\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}}$

This ratio represents a rotation in the complex plane, and it corresponds to a constant angular difference between consecutive roots.

Tip:

When solving for the roots of complex numbers, remember to use De Moivre's Theorem, and always express complex numbers in polar form before solving for the roots.

Would you like further details on any part of the solution, or have any other questions? Here are some related questions for you:

1. How do you apply De Moivre's Theorem to find nth roots of a complex number?
2. What are the steps for converting a complex number from Cartesian form to polar form?
3. How can you verify the accuracy of your roots when solving complex equations?
4. What does it mean for the roots of an equation to form a geometric sequence?
5. How does the argument of a complex number affect its positioning on the complex plane?

Let me know if you need further clarification!