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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.

Learn how to find the fourth roots of the imaginary unit i using De Moivre's Theorem. This solution covers complex numbers, their polar form, and applying De Moivre’s theorem for solving higher-order roots. Suitable for undergraduate students studying complex numbers or advanced algebra.

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