Problem 90 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The fifth roots of -16 - 16\sqrt{3}i are 2 \text{cis}(\frac{4\pi}{15}), 2 \text{cis}(\frac{10\pi}{15}), 2 \text{cis}(\frac{16\pi}{15}), 2 \text{cis}(\frac{22\pi}{15}), 2 \text{cis}(\frac{28\pi}{15}) , and they form a pentagon on the complex plane.

Step 1: Express the Complex Number in Polar Form

The given complex number is \(-16 - 16 \sqrt{3} i\). To express it in polar form, we need the magnitude \( r \) and the angle \( \theta \).Calculate the magnitude \( r \):\[ r = \sqrt{(-16)^2 + (-16 \sqrt{3})^2} = \sqrt{256 + 768} = 32 \] Calculate the angle \( \theta \) using the tangent function:\[ \theta = \tan^{-1}\left(\frac{-16 \sqrt{3}}{-16}\right) = \tan^{-1}(\sqrt{3}) = 240^\circ \text{ or } \pi + \frac{\pi}{3} = \frac{4\pi}{3} \] Thus, the polar form is \( 32 \text{cis} \frac{4\pi}{3} \).

Step 2: Find the Fifth Roots Using De Moivre's Theorem

According to De Moivre's theorem, if \( z = r \text{cis} \theta \), the \( n^{th} \) roots of \( z \) are given by \( \sqrt[n]{r} \text{cis} \left( \frac{\theta + 2k\pi}{n} \right) \), where \( k = 0, 1, 2, \, ..., \, n-1 \).For the fifth roots of \( 32 \text{cis} \frac{4\pi}{3} \), we have \( n = 5 \). Thus, the roots are:\[ \sqrt[5]{32} = 2 \]Each root is: \[ 2 \text{cis} \left( \frac{\frac{4\pi}{3} + 2k\pi}{5} \right) \text{ for } k = 0, 1, 2, 3, 4 \]

Step 3: Calculate Each Root

Calculate the angles for each value of \( k \):\( k = 0 \): \[ 2 \text{cis} \left( \frac{4\pi}{15} \right) \] \( k = 1 \): \[ 2 \text{cis} \left( \frac{4\pi}{15} + \frac{2\pi}{5} \right) = 2 \text{cis} \left( \frac{10\pi}{15} \right) \] \( k = 2 \): \[ 2 \text{cis} \left( \frac{4\pi}{15} + \frac{4\pi}{5} \right) = 2 \text{cis} \left( \frac{16\pi}{15} \right) \] \( k = 3 \): \[ 2 \text{cis} \left( \frac{4\pi}{15} + \frac{6\pi}{5} \right) = 2 \text{cis} \left( \frac{22\pi}{15} \right) \] \( k = 4 \): \[ 2 \text{cis} \left( \frac{4\pi}{15} + \frac{8\pi}{5} \right) = 2 \text{cis} \left( \frac{28\pi}{15} \right) \]

Step 4: Graph the Roots

To graph these roots in the complex plane, remember that each root \((r \text{cis} \theta)\) can be represented by the complex number \( r(\cos \theta + i\sin \theta) \).These points are at equal intervals on a circle of radius 2, centered at the origin. Plot the points corresponding to: (1) \( 2 \) units at angle \( \frac{4\pi}{15} \), (2) \( 2 \) units at angle \( \frac{10\pi}{15} \), (3) \( 2 \) units at angle \( \frac{16\pi}{15} \), (4) \( 2 \) units at angle \( \frac{22\pi}{15} \), (5) \( 2 \) units at angle \( \frac{28\pi}{15} \).

Step 5: Verify the Results

Ensure all roots are distinct and equidistant on the circle, confirming they are separated by \( \frac{2\pi}{5} \) radians. This symmetry verifies correct application of De Moivre's Theorem.

Key Concepts

Polar Form

The polar form of a complex number is a compact way to represent it using a magnitude and an angle, rather than traditional rectangular coordinates. This makes it particularly useful for multiplication, division, and finding powers and roots of complex numbers. In polar form, a complex number is expressed as \( r \text{cis} \theta \) or equivalently, \( r(\cos \theta + i\sin \theta) \).
Here, \( r \) stands for the magnitude or modulus, which is the distance from the origin to the point in the complex plane. It is calculated as \( \sqrt{x^2 + y^2} \) for a complex number given in rectangular form \( x + yi \).
The angle \( \theta \), also known as the argument, is measured in radians or degrees and is the angle formed with the positive x-axis.

• Magnitude: Gives the "size" of the complex number.
• Argument: Provides the direction of the complex number in the plane.

Polar form becomes advantageous with operations that involve exponents or roots, as they are more naturally dealt with angles and magnitudes.
For example, the conversion of \(-16 - 16\sqrt{3} i\) into polar form was performed by finding \( r = 32 \) and \( \theta = \frac{4\pi}{3} \). This polar form simplifies the calculation of powers and roots.

De Moivre's Theorem

De Moivre's Theorem is an invaluable tool in complex number arithmetic, especially for finding roots and powers. It states that for any complex number in polar form \( z = r \text{cis} \theta \) and integer \( n \), the power \( z^n \) can be obtained by raising the magnitude to the power of \( n \) and multiplying the angle by \( n \). This can be expressed mathematically as:
\[ z^n = r^n \text{cis}(n\theta) \]
When finding roots, the theorem is adapted to:
\[ \sqrt[n]{z} = \sqrt[n]{r} \text{cis} \left(\frac{\theta + 2k\pi}{n}\right) \text{ for } k = 0, 1, 2, \ldots, n-1 \]

• The theorem breaks down the complex calculations into simple arithmetic operations.
• It helps identify all possible roots systematically, spaced equally in the complex plane.

For example, to find the fifth roots of \( 32 \text{cis} \frac{4\pi}{3} \), De Moivre's Theorem helps predict exactly where each root will lie: separated by angles of \( \frac{2\pi}{5} \) radians.
This property is crucial for graphing and understanding the symmetrical nature of complex roots.

Complex Plane

The complex plane, also known as the Argand plane, is a two-dimensional representation of complex numbers. This plane allows us to visually interpret complex numbers, using a horizontal real axis and a vertical imaginary axis.

• Real numbers are represented along the horizontal axis, denoted as the "real axis."
• Imaginary numbers appear along the vertical axis, known as the "imaginary axis."

In this coordinate system, each complex number \( x + yi \) maps directly to a point \((x, y)\). Understanding these coordinates is essential when working with operations and transformations in the complex domain.
When we graph complex roots, like the fifth roots of \(-16-16\sqrt{3} i\), they appear as points spaced symmetrically around a circle at a radius equal to the magnitude of the roots.
Visualizing these positions helps confirm calculations and understand how complex operations affect numbers. Thus, familiarity with plotting on the complex plane is key to mastering complex number manipulation.