Math Problem Statement

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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Solution

To find the fourth roots of the complex number 6+63i6 + 6\sqrt{3} i, we can express it in polar form and then use De Moivre's theorem.

Step 1: Convert 6+63i6 + 6\sqrt{3} i to Polar Form

  1. Calculate the modulus: r=(6)2+(63)2=36+108=144=12r = \sqrt{(6)^2 + (6\sqrt{3})^2} = \sqrt{36 + 108} = \sqrt{144} = 12

  2. Find the argument θ\theta: θ=tan⁡−1(636)=tan⁡−1(3)=π3\theta = \tan^{-1}\left(\frac{6\sqrt{3}}{6}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}

So, in polar form, 6+63i6 + 6\sqrt{3} i can be written as: 12cisπ3or12(cos⁡π3+isin⁡π3)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⁡+isin⁡\cos + i \sin.

Step 2: Apply De Moivre's Theorem for the Fourth Roots

To find the fourth roots, we need to compute: zk=121/4cis(π3+2kπ4)fork=0,1,2,3z_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: 121/4=124=2312^{1/4} = \sqrt[4]{12} = \sqrt{2 \sqrt{3}}

  2. Calculate the arguments for each root:

    • For k=0k = 0: θ0=π12=π3/4\theta_0 = \frac{\pi}{12} = \frac{\pi}{3} / 4

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polar Form
Roots of Complex Numbers

Formulas

Modulus of a complex number: r = √(a^2 + b^2)
Argument of a complex number: θ = tan⁻¹(b/a)
De Moivre's Theorem: (r cis θ)^n = r^n cis(nθ)

Theorems

De Moivre's Theorem

Suitable Grade Level

College Level

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