The fifth roots of -16 -1613i
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Find the indicated roots and graph them in complex plane
![**Title: Finding the Fifth Roots of a Complex Number**
**Content:**
In complex number theory, one can find multiple roots for a given complex number. Here, we explore how to find the fifth roots of the complex number \(-16 - 16\sqrt{3}i\).
**Steps to Solve:**
1. **Convert to Polar Form:**
- Identify the modulus \(r\) of the complex number, calculated as \(|z| = \sqrt{(-16)^2 + (-16\sqrt{3})^2}\).
- Determine the argument \(\theta\) using \(\tan\theta = \frac{\text{Imaginary part}}{\text{Real part}}\).
2. **Apply De Moivre's Theorem:**
- The \(n^{th}\) roots of a complex number \(z = r(\cos\theta + i\sin\theta)\) can be found using the formula:
\[
z_k = r^{1/n} \left( \cos\left( \frac{\theta + 2k\pi}{n} \right) + i\sin\left( \frac{\theta + 2k\pi}{n} \right) \right)
\]
- For \(n = 5\), compute all roots \(z_0, z_1, z_2, z_3, z_4\) by substituting \(k = 0, 1, 2, 3, 4\).
This approach helps visualize complex numbers and understand their behavior in the complex plane, specifically when finding their roots.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F35a669ef-8ceb-4476-b34d-cf3aac0d1454%2F932a4baa-f8a6-4d49-b4cd-22b661131767%2Ftuvhdly.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Fifth Roots of a Complex Number**
**Content:**
In complex number theory, one can find multiple roots for a given complex number. Here, we explore how to find the fifth roots of the complex number \(-16 - 16\sqrt{3}i\).
**Steps to Solve:**
1. **Convert to Polar Form:**
- Identify the modulus \(r\) of the complex number, calculated as \(|z| = \sqrt{(-16)^2 + (-16\sqrt{3})^2}\).
- Determine the argument \(\theta\) using \(\tan\theta = \frac{\text{Imaginary part}}{\text{Real part}}\).
2. **Apply De Moivre's Theorem:**
- The \(n^{th}\) roots of a complex number \(z = r(\cos\theta + i\sin\theta)\) can be found using the formula:
\[
z_k = r^{1/n} \left( \cos\left( \frac{\theta + 2k\pi}{n} \right) + i\sin\left( \frac{\theta + 2k\pi}{n} \right) \right)
\]
- For \(n = 5\), compute all roots \(z_0, z_1, z_2, z_3, z_4\) by substituting \(k = 0, 1, 2, 3, 4\).
This approach helps visualize complex numbers and understand their behavior in the complex plane, specifically when finding their roots.
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