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Magazine Chapter 7: Problem 33
Find all \(n\) th roots of \(z\). Write the answers in polar form, and plot the roots in the complex plane. $$\sqrt{18}-\sqrt{18} i, n=2$$
Short Answer
Expert verified
The square roots of \( \sqrt{18} - \sqrt{18}i \) are \( \sqrt{6}\text{cis}(-\frac{\pi}{8}) \) and \( \sqrt{6}\text{cis}(\frac{7\pi}{8}) \).
Step by step solution
01
Convert the Complex Number to Polar Form
The given complex number is \( \sqrt{18} - \sqrt{18}i \). To convert it to polar form, we need the modulus \( r \) and the argument \( \theta \). The modulus is calculated as \( r = \sqrt{x^2 + y^2} \), where \( x = \sqrt{18} \) and \( y = -\sqrt{18} \). Thus, \( r = \sqrt{(\sqrt{18})^2 + (-\sqrt{18})^2} = \sqrt{18 + 18} = 6 \). The argument is given by \( \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-\sqrt{18}}{\sqrt{18}}\right) = \tan^{-1}(-1) = -\frac{\pi}{4} \). Therefore, the polar form is \( z = 6\text{cis}(-\frac{\pi}{4}) \).
02
Find the Principal Root
To find the square roots of a complex number, we use the formula \( z^{1/n} = r^{1/n}\text{cis}\left(\frac{\theta + 2k\pi}{n}\right) \) for each integer \( k \). Here, \( r = 6 \), \( \theta = -\frac{\pi}{4} \), and \( n = 2 \). The principal root corresponds to \( k = 0 \):\[ z_0 = 6^{1/2}\text{cis}\left(\frac{-\pi/4}{2}\right) = \sqrt{6}\text{cis}\left(-\frac{\pi}{8}\right) \].
03
Find the Second Root
To find the second root, use the formula from Step 2 with \( k = 1 \):\[ z_1 = 6^{1/2}\text{cis}\left(\frac{-\pi/4 + 2\pi}{2}\right) = \sqrt{6}\text{cis}\left(\frac{7\pi}{8}\right) \].
04
Plot the Roots in the Complex Plane
In the complex plane, each root has a magnitude of \( \sqrt{6} \). The angles to plot are \( -\frac{\pi}{8} \) for the principal root and \( \frac{7\pi}{8} \) for the second root. In polar coordinates, these respectively correspond to the points:1. \( \sqrt{6}\text{cis}\left(-\frac{\pi}{8}\right) \) - located just below the x-axis in the first quadrant.2. \( \sqrt{6}\text{cis}\left(\frac{7\pi}{8}\right) \) - located just above the x-axis in the second quadrant, leftward.
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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 of expressing complex numbers using a combination of modulus and angle. When you have a number like \( z = a + bi \), converting it to polar form involves calculating two things:
For the given complex number, \( \sqrt{18} - \sqrt{18}i \), converting to polar form involves calculating \( r = 6 \) and \( \theta = -\frac{\pi}{4} \), resulting in \( z = 6\text{cis}(-\frac{\pi}{4}) \).
- Modulus \( r \): This is the distance from the origin to the number in the complex plane. It's found using the formula \( r = \sqrt{a^2 + b^2} \).
- Argument \( \theta \): This is the angle formed with the positive x-axis. You can find it using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).
For the given complex number, \( \sqrt{18} - \sqrt{18}i \), converting to polar form involves calculating \( r = 6 \) and \( \theta = -\frac{\pi}{4} \), resulting in \( z = 6\text{cis}(-\frac{\pi}{4}) \).
n-th Roots
Finding the \( n \)-th roots of a complex number means finding all the numbers that, when raised to the power of \( n \), give the original complex number. The formula used is:
Here are the steps to follow:
- \( z^{1/n} = r^{1/n}\text{cis}\left(\frac{\theta + 2k\pi}{n}\right) \)
Here are the steps to follow:
- Calculate \( r^{1/2} \): This gives the magnitude of the roots, in our case, \( \sqrt{6} \).
- Find each root: Start with \( k = 0 \) for the principal root, resulting in \( \sqrt{6}\text{cis}(-\frac{\pi}{8}) \). Use \( k = 1 \) for the second root, leading to \( \sqrt{6}\text{cis}(\frac{7\pi}{8}) \).
Complex Plane
The complex plane is a way of graphically representing complex numbers. It involves a two-dimensional grid where:
- The real part of the number is plotted on the x-axis.
- The imaginary part of the number is plotted on the y-axis.
- First root \( \sqrt{6}\text{cis}(-\frac{\pi}{8}) \): Plotted below the x-axis in the first quadrant.
- Second root \( \sqrt{6}\text{cis}(\frac{7\pi}{8}) \): Plotted above the x-axis in the second quadrant.
Modulus and Argument
The modulus and argument are essential concepts when dealing with complex numbers, particularly in polar form.
- Modulus \( r \): Represents the magnitude or absolute value of the complex number, calculated with \( r = \sqrt{a^2 + b^2} \). It measures how far from the origin the number is located in the complex plane.
- Argument \( \theta \): This is the angle between the positive x-axis and the line joining the origin with the number in the complex plane. Calculated with \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \), it helps in defining the direction of the number.
