https://file.aiquickdraw.com/mathbot/file/17317386873305fwp5m7u.jpg

To solve for the complex cube roots of \( 2 + 2i \), we proceed as follows:

### Step 1: Convert to polar form
The complex number \( 2 + 2i \) can be written in polar form as:
\[
z = r e^{i\theta}
\]
where:
\[
r = |z| = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)
\]

Here, \( x = 2 \) and \( y = 2 \):
\[
r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}
\]

The angle \( \theta \) is:
\[
\theta = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = \frac{\pi}{4}
\]

Thus, the polar form of \( z \) is:
\[
z = 2\sqrt{2} e^{i\frac{\pi}{4}}
\]

---

### Step 2: Find the cube roots
The formula for the \( n \)-th roots of a complex number is:
\[
z_k = r^{1/n} e^{i(\theta + 2k\pi)/n}, \quad k = 0, 1, 2, \dots, n-1
\]

Here, \( n = 3 \), so:
\[
r^{1/3} = (2\sqrt{2})^{1/3}, \quad \text{and} \quad z_k = (2\sqrt{2})^{1/3} e^{i(\frac{\pi}{4} + \frac{2k\pi}{3})}, \quad k = 0, 1, 2
\]

---

### Step 3: Write the roots
1. For \( k = 0 \):
\[
z_0 = (2\sqrt{2})^{1/3} e^{i\frac{\pi}{12}}
\]

2. For \( k = 1 \):
\[
z_1 = (2\sqrt{2})^{1/3} e^{i\left(\frac{\pi}{4} + \frac{2\pi}{3}\right)} = (2\sqrt{2})^{1/3} e^{i\frac{11\pi}{12}}
\]

3. For \( k = 2 \):
\[
z_2 = (2\sqrt{2})^{1/3} e^{i\left(\frac{\pi}{4} + \frac{4\pi}{3}\right)} = (2\sqrt{2})^{1/3} e^{i\frac{19\pi}{12}}
\]

---

### Final Answer
The complex cube roots of \( 2 + 2i \) are:
\[
(2\sqrt{2})^{1/3} e^{i\frac{\pi}{12}}, \quad (2\sqrt{2})^{1/3} e^{i\frac{11\pi}{12}}, \quad (2\sqrt{2})^{1/3} e^{i\frac{19\pi}{12}}
\]

Would you like me to further simplify or convert these results? Let me know!

---

### Related Questions:
1. What is the geometric representation of the roots on the complex plane?
2. How can we express these roots in rectangular (Cartesian) form?
3. What is the magnitude of each cube root?
4. How do cube roots differ for different complex numbers?
5. Can this method be used for higher-order roots (e.g., 4th roots)?

---

**Tip:** Always convert the complex number into polar form for efficient computation of roots or powers!

Find all the complex cube roots of 2 + 2i. Write the answer in exponential form.

Learn how to find the complex cube roots of 2 + 2i by converting the number into polar form and applying the roots of a complex number formula. This detailed solution includes step-by-step calculations, angle simplifications, and exponential form representation. Ideal for advanced high school or early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation