
**Homework:10.3 HW - The Complex Plaine; De Moivre's Theorem**

**Question 25,** **10.3.59**

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Part 1

Find all the complex roots. Write the answer in exponential form and in polar form.

The complex fourth roots of 6 plus 6 StartRoot 3 EndRoot i6+63i

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Part 1

The complex fourth roots in exponential form are 

enter your response here.

​(Simplify your answers. Type exact​ answers, using 

piπ 

as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than 

2piπ. 

Use integers or fractions for any numbers in the expressions. Use a comma to separate answers as​ needed.) 

The complex fourth roots in exponential form 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 expressions. Use a comma to separate answers as needed.)

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To find the fourth roots of the complex number \( 6 + 6\sqrt{3} i \), we can express it in polar form and then use De Moivre's theorem.

### Step 1: Convert \( 6 + 6\sqrt{3} i \) to Polar Form
1. **Calculate the modulus**:
   \[
   r = \sqrt{(6)^2 + (6\sqrt{3})^2} = \sqrt{36 + 108} = \sqrt{144} = 12
   \]

2. **Find the argument** \(\theta\):
   \[
   \theta = \tan^{-1}\left(\frac{6\sqrt{3}}{6}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}
   \]

So, in polar form, \( 6 + 6\sqrt{3} i \) can be written as:
\[
12 \text{cis} \frac{\pi}{3} \quad \text{or} \quad 12 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)
\]
where "cis" represents \( \cos + i \sin \).

### Step 2: Apply De Moivre's Theorem for the Fourth Roots
To find the fourth roots, we need to compute:
\[
z_k = 12^{1/4} \text{cis} \left(\frac{\frac{\pi}{3} + 2k\pi}{4}\right) \quad \text{for} \quad k = 0, 1, 2, 3
\]

1. **Calculate the modulus of the root**:
   \[
   12^{1/4} = \sqrt[4]{12} = \sqrt{2 \sqrt{3}}
   \]

2. **Calculate the arguments for each root**:
   - For \( k = 0 \):
     \[
     \theta_0 = \frac{\pi}{12} = \frac{\pi}{3} / 4
     \]


Find all the complex roots. Write the answer in exponential form and in polar form. The complex fourth roots of 6 + 6√3 i.

Discover how to find the fourth roots of the complex number 6 + 6√3 i, expressed in both exponential and polar forms. This solution uses key complex number concepts, including modulus, argument, and De Moivre's Theorem. Ideal for college-level math students.

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