Problem 78 Find the indicated roots, and gr... [FREE SOLUTION]

Chapter 8: Problem 78

Find the indicated roots, and graph the roots in the complex plane. The cube roots of \(4 \sqrt{3}+4 i\)

Short Answer

Expert verified

The cube roots are approximately \(1.98+0.35i\), \(-1.98+0.35i\), and \(0-2i\). Graph these on the complex plane.

Step by step solution

Convert to Polar Form

To find the cube roots of a complex number, first express it in polar form. The complex number is given as \(4 \sqrt{3} + 4i\). Find the modulus: \(r = \sqrt{(4\sqrt{3})^2 + 4^2} = \sqrt{48 + 16} = \sqrt{64} = 8\). Next, find the argument \( \theta \) using \(\tan \theta = \frac{4}{4\sqrt{3}} = \frac{1}{\sqrt{3}}\). Hence, \( \theta = \frac{\pi}{6}\). The polar form is \( 8 \text{cis} \frac{\pi}{6} \).

Find Cube Roots

The cube roots of a complex number in polar form \( r \text{cis} \theta \) are given by \( \sqrt[3]{r} \text{cis} \left( \frac{\theta + 2k\pi}{3} \right) \) for \( k = 0, 1, 2 \). The cube root of the modulus is \( \sqrt[3]{8} = 2 \). The three roots are:- \( 2 \text{cis} \left( \frac{\pi}{18} \right) \) for \( k = 0\),- \( 2 \text{cis} \left( \frac{\pi}{18} + \frac{2\pi}{3} \right) = 2 \text{cis} \left( \frac{13\pi}{18} \right) \) for \( k = 1\),- \( 2 \text{cis} \left( \frac{\pi}{18} + \frac{4\pi}{3} \right) = 2 \text{cis} \left( \frac{25\pi}{18} \right) \) for \( k = 2\).

Convert Back to Rectangular Form

Convert each of the roots from polar back to rectangular form using \( \text{cis} \theta = \cos \theta + i \sin \theta \). The roots become:- For \( 2 \text{cis} \left( \frac{\pi}{18} \right) \), it's approximately \( 2 \left( \cos \frac{\pi}{18} + i \sin \frac{\pi}{18} \right) \approx 1.98 + 0.35i \).- For \( 2 \text{cis} \left( \frac{13\pi}{18} \right) \), it's approximately \( 2 \left( \cos \frac{13\pi}{18} + i \sin \frac{13\pi}{18} \right) \approx -1.98 + 0.35i \).- For \( 2 \text{cis} \left( \frac{25\pi}{18} \right) \), it's approximately \( 2 \left( \cos \frac{25\pi}{18} + i \sin \frac{25\pi}{18} \right) \approx 0 - 2i \).

Graph the Roots

To graph the roots on the complex plane, place each root at its respective coordinate:- \( (1.98, 0.35) \) for the first root.- \( (-1.98, 0.35) \) for the second root.- \( (0, -2) \) for the third root.These points form an equilateral triangle centered around the origin on the complex plane.

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Key Concepts

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

Polar Form

Polar form is a way to represent complex numbers, which can make it easier to perform certain calculations. Rather than writing a complex number as \( a + bi \) (called rectangular form), it can be expressed as \( r \text{cis} \theta \), where:

\( r \) is the modulus or absolute value of the complex number.
\( \theta \) is the argument, representing the angle formed with the positive x-axis in the complex plane.
\( \text{cis} \theta \) is a shorthand for \( \cos \theta + i \sin \theta \).

To convert from rectangular form to polar form, you calculate the modulus with \( r = \sqrt{a^2 + b^2} \), and the argument \( \theta \) through \( \tan \theta = \frac{b}{a} \). This representation is especially useful when multiplying or dividing complex numbers, as the operations become simpler.

Cube Roots

Finding cube roots of a complex number involves expressing the number in polar form first. When a complex number is expressed as \( r \text{cis} \theta \), its cube roots are found by using the formula:

\( \sqrt[3]{r} \text{cis}\left( \frac{\theta + 2k\pi}{3} \right) \)

where \( k = 0, 1, \) or \( 2 \), as there are three distinct cube roots. Each value of \( k \) provides a different cube root.This formula takes advantage of the polar form representation to easily calculate roots by dealing primarily with the angle, adjusting it by \( 2k\pi \) to get multiple roots. In simpler terms, it breaks down complex multiplications and divisions into simpler arithmetic operations of angles.

Complex Plane

The complex plane provides a visual representation of complex numbers. Imagine a two-dimensional plane where:

The horizontal axis (x-axis) represents real numbers.
The vertical axis (y-axis) represents imaginary numbers.

Each complex number corresponds to a point on this plane with coordinates \( (a, b) \) for the number \( a + bi \). Graphing the roots of a complex number on this plane can reveal symmetries and patterns, like how the cube roots often form geometric shapes such as an equilateral triangle centered around the origin. Understanding the layout of the complex plane is fundamental for visualizing the effects of complex arithmetic.

Rectangular Form

Rectangular form expresses a complex number as \( a + bi \), where:

\( a \) is the real part.
\( b \) is the imaginary part.

This form is very useful when adding or subtracting complex numbers, as it allows straightforward arithmetic operations directly with the real and imaginary parts.Converting back from polar to rectangular form involves using the expressions for cosine and sine. Specifically, given \( r \text{cis} \theta \), the rectangular form is obtained by calculating \( a = r \cos \theta \) and \( b = r \sin \theta \). This backwards conversion is critical when easing back into practical or numerical interpretations after performing complex arithmetic operations in polar form.

91影视

Find the indicated roots, and graph the roots in the complex plane. The cube roots of \(4 \sqrt{3}+4 i\)

Short Answer

Step by step solution

Convert to Polar Form

Find Cube Roots

Convert Back to Rectangular Form

Graph the Roots

Key Concepts

Polar Form

Cube Roots

Complex Plane

Rectangular Form

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