Problem 86 Find the indicated roots, and gr... [FREE SOLUTION]

Chapter 9: Problem 86

Find the indicated roots, and graph the roots in the complex plane. The fifth roots of \(-16-16 \sqrt{3} i\)

Short Answer

Expert verified

The fifth roots are 2cis(4蟺/15), 2cis(10蟺/15), 2cis(16蟺/15), 2cis(22蟺/15), and 2cis(28蟺/15).

Step by step solution

Express Complex Number in Polar Form

First, express the complex number \(-16 - 16\sqrt{3}i\) in polar form. To do this, we need to find the magnitude \(r\) and the argument \(\theta\). The magnitude is calculated as \(r = \sqrt{(-16)^2 + (-16\sqrt{3})^2} = \sqrt{256 + 768} = 32\). To find \(\theta\), use the tangent function: \(\tan\theta = \frac{\text{Imaginary part}}{\text{Real part}} = \frac{-16\sqrt{3}}{-16} = \sqrt{3}\), which means \(\theta = \frac{4\pi}{3}\) (in the third quadrant). Therefore, in polar form it is \(32\text{cis}\frac{4\pi}{3}\).

Apply De Moivre's Theorem for Roots

To find the fifth roots, apply De Moivre's Theorem. The roots are given by:\[ z_k = 32^{1/5} \text{cis} \left( \frac{\frac{4\pi}{3} + 2k\pi}{5} \right) \] for \(k = 0, 1, 2, 3, 4\). Here, \(32^{1/5} = 2\).

Calculate Each Root

Calculate the argument for each root:- For \(k = 0\), \(z_0 = 2\text{cis} \left( \frac{4\pi}{15} \right) \)- For \(k = 1\), \(z_1 = 2\text{cis} \left( \frac{10\pi}{15} \right) \)- For \(k = 2\), \(z_2 = 2\text{cis} \left( \frac{16\pi}{15} \right) \)- For \(k = 3\), \(z_3 = 2\text{cis} \left( \frac{22\pi}{15} \right) \)- For \(k = 4\), \(z_4 = 2\text{cis} \left( \frac{28\pi}{15} \right) \)

Graph the Roots in the Complex Plane

Plot each of the roots extracted from polar form, using the magnitudes and arguments calculated in Step 3. Each root is a point in the complex plane at distance 2 from the origin, with angles corresponding to their calculated arguments. The points will be equally distributed around a circle of radius 2, centered at the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

De Moivre's Theorem

De Moivre's Theorem is a powerful formula used in complex number operations, especially when raising complex numbers to powers or finding roots. It states:

\[ (r \text{cis} \theta)^n = r^n \text{cis} (n\theta) \]
Here, "cis" stands for \(\cos \theta + i \sin \theta\), which is a shorthand used in trigonometry for ease. When dealing with roots, De Moivre's Theorem allows us to find multiple solutions by taking the \(n\)-th root of the magnitude \(r\) and adjusting the angle \(\theta\) accordingly.

- To find the \(n\)-th roots using De Moivre's Theorem:

Determine \(r^{1/n}\), the \(n\)-th root of the magnitude.
Calculate the angle: divide the original angle by \(n\) and consider adding \(\frac{2k\pi}{n}\) for \(k = 0, 1, ..., n-1\).

This ensures all possible roots are captured, showing the symmetry and periodic nature of roots in the complex plane.

Polar Form

The polar form of a complex number provides a fascinating way to understand and manipulate it, especially when using trigonometric identities and performing multiplications or divisions. A complex number \(a + bi\) can be transformed into polar form as \(r \text{cis} \theta\).

- To convert to polar form:

Find the magnitude \(r\) using \(\sqrt{a^2 + b^2}\).
Determine the angle \(\theta\) using \(\tan \theta = \frac{b}{a}\).

Polar form is particularly useful because it simplifies multiplication and division through the properties of angles and magnitudes. This form highlights the rotation (angle) and scaling (magnitude) properties intrinsic to complex numbers.

Complex Plane Graphing

Graphing in the complex plane offers a visual insight into the nature and behavior of complex numbers. The complex plane is a two-dimensional plane where each point represents a complex number. The horizontal axis (real axis) corresponds to the real part, and the vertical axis (imaginary axis) corresponds to the imaginary part.

- Key points when graphing:

Each complex number is a point: \((a, b)\) for \(a + bi\).
Roots of an equation or expression can create symmetrical patterns such as polygons.
The angle from the positive real axis, heading counter-clockwise, represents the argument of the number.

Understanding these graphical representations helps in visualizing operations such as rotations (adding angles) and scaling (changing magnitudes), making complex number operations much intuitive.

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91影视

Find the indicated roots, and graph the roots in the complex plane. The fifth roots of \(-16-16 \sqrt{3} i\)

Short Answer

Step by step solution

Express Complex Number in Polar Form

Apply De Moivre's Theorem for Roots

Calculate Each Root

Graph the Roots in the Complex Plane

Key Concepts

De Moivre's Theorem

Polar Form

Complex Plane Graphing

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