
Homework:10.3
Question 14, 10.3.57
HW Score: 81.25%, 13 of 16 points
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Part 1
Find all the complex roots. Write the answer in exponential form.
The complex cube roots of 3 plus 3 i.
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Part 1
The complex cube roots are 
  
enter your response here.
​(Simplify your answers. Type exact​ answers, using pi as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than 2pi. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.) 
The complex cube roots are(Simplify your answers. Type exact answers, using as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than 2. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
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We are tasked with finding the complex cube roots of \( 3 + 3i \) and expressing the answers in exponential form. To do this, we'll follow these steps:

### Step 1: Convert the complex number \( 3 + 3i \) to polar form.

The polar form of a complex number \( z = x + yi \) is written as:
\[
z = r \cdot e^{i\theta}
\]
where:
- \( r = \sqrt{x^2 + y^2} \) is the modulus (magnitude) of the complex number,
- \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \) is the argument (angle).

For \( 3 + 3i \), we calculate:
- \( r = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \),
- \( \theta = \tan^{-1}\left(\frac{3}{3}\right) = \tan^{-1}(1) = \frac{\pi}{4} \).

So, the polar form of \( 3 + 3i \) is:
\[
3 + 3i = 3\sqrt{2} \cdot e^{i\frac{\pi}{4}}.
\]

### Step 2: Find the cube roots.

The formula for the \( n \)-th roots of a complex number \( z = r \cdot e^{i\theta} \) is:
\[
z_k = \sqrt[n]{r} \cdot e^{i\left(\frac{\theta + 2k\pi}{n}\right)}
\]
for \( k = 0, 1, 2, \dots, n-1 \).

For cube roots (\( n = 3 \)), we apply the formula:
- \( r = 3\sqrt{2} \), so \( \sqrt[3]{r} = \sqrt[3]{3\sqrt{2}} = \left(3\sqrt{2}\right)^{1/3} \),
- The argument \( \theta = \frac{\pi}{4} \), and we will calculate the three cube roots for \( k = 0, 1, 2 \).

#### Root 1: \( k = 0 \)
\[
z_0 = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\frac{\pi}{4} + 2(0)\pi}{3}} = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\pi}{12}}.
\]

#### Root 2: \( k = 1 \)
\[
z_1 = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\frac{\pi}{4} + 2(1)\pi}{3}} = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{7\pi}{12}}.
\]

#### Root 3: \( k = 2 \)
\[
z_2 = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\frac{\pi}{4} + 2(2)\pi}{3}} = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{13\pi}{12}}.
\]

### Final Answer
The complex cube roots of \( 3 + 3i \) are:
\[
\left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\pi}{12}}, \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{7\pi}{12}}, \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{13\pi}{12}}.
\]

Would you like to explore more details or have further questions?

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Here are 5 related questions you can try:
1. What are the complex cube roots of \( 8 + 8i \)?
2. Convert the complex number \( 4 + 4i \) to polar form.
3. How do you find the fourth roots of a complex number?
4. What is the general formula for finding the \( n \)-th roots of a complex number?
5. Can you explain how the modulus and argument relate to the polar form of a complex number?

### Tip:
Always ensure your angle \( \theta \) is in the correct range (between 0 and \( 2\pi \)) when writing complex roots in exponential form!

Find all the complex cube roots of 3 + 3i. Write the answer in exponential form.

Learn how to find the complex cube roots of 3 + 3i using polar and exponential forms. This step-by-step guide covers converting complex numbers to polar form and applying the cube root formula, suitable for Grades 11-12.

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