The image contains the handwritten phrase: "The fourth roots of $i$ = ±1, ±i."

This statement refers to finding the fourth roots of the complex number $i$, where $i$ is the imaginary unit, defined as $i^2 = -1$.

Solving for the Fourth Roots of $i$:

The fourth roots of a complex number can be found using polar coordinates and De Moivre's Theorem. Let’s break it down:

1. Express $i$ in polar form:

   • $i = 0 + 1i$, which in polar form is: $r = 1 \quad \text{(magnitude)}, \quad \theta = \frac{\pi}{2} \quad \text{(angle, or argument)}$ Hence, $i = \text{cis}\left( \frac{\pi}{2} \right)$, where $\text{cis}(\theta)$ is shorthand for $\cos(\theta) + i \sin(\theta)$.

2. Apply De Moivre's Theorem:

   • To find the fourth roots, we need to solve $z^4 = i$. According to De Moivre’s Theorem: $z_k = \sqrt[4]{r} \cdot \text{cis}\left( \frac{\theta + 2k\pi}{4} \right), \quad k = 0, 1, 2, 3$ Since $r = 1$, $\sqrt[4]{r} = 1$, and $\theta = \frac{\pi}{2}$, we get: $z_k = \text{cis} \left( \frac{\frac{\pi}{2} + 2k\pi}{4} \right)$

3. Find the four distinct roots (for $k = 0, 1, 2, 3$):

   • For $k = 0$: $z_0 = \text{cis} \left( \frac{\pi}{8} \right) = \cos\left( \frac{\pi}{8} \right) + i\sin\left( \frac{\pi}{8} \right)$
   • For $k = 1$: $z_1 = \text{cis} \left( \frac{5\pi}{8} \right) = \cos\left( \frac{5\pi}{8} \right) + i\sin\left( \frac{5\pi}{8} \right)$
   • For $k = 2$: $z_2 = \text{cis} \left( \frac{9\pi}{8} \right) = \cos\left( \frac{9\pi}{8} \right) + i\sin\left( \frac{9\pi}{8} \right)$
   • For $k = 3$: $z_3 = \text{cis} \left( \frac{13\pi}{8} \right) = \cos\left( \frac{13\pi}{8} \right) + i\sin\left( \frac{13\pi}{8} \right)$

These give the four fourth roots of $i$. The statement suggests that two of them are $\pm 1$ and the other two are $\pm i$, but the exact values are more complex when expressed in trigonometric form.

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

Here are 5 related questions to explore:

1. What is De Moivre's Theorem and how is it applied?
2. How do you convert complex numbers into polar form?
3. What are other methods to find the roots of a complex number?
4. Can we generalize this process to find any $n$-th root of a complex number?
5. How do you verify that these four values satisfy $z^4 = i$?

Tip: When dealing with complex numbers, always consider expressing them in polar or exponential form for easier manipulation with powers and roots.