Q. 64 Show that complex nth root of a ... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 64 (page 592)

Show that complex nth root of a nonzero complex number w lies on a circle with centre at the origin. What is the radius of this circle?

Short Answer

Expert verified

The points form a circle with centre at the origin and radius equals to $\sqrt[m]{r}$

Step by step solution

Step 1. Considering a complex number w

Assume $w = r (\cos 胃_{鈭�} + i \sin 胃_{鈭�})$ be a non zero complex number. According to the theorem it has m distinct roots where $m 鈮� 2$

$z_{n} = \sqrt[m]{r} [(\cos (\frac{胃_{慰}}{m} + \frac{2 n 蟺}{m}) + i \sin (\frac{胃_{慰}}{m} + \frac{2 n 蟺}{m}))]$

n=0,1,2,3........(m-1)

Step 2. Calculations

Now if we plot this complex roots on a graph with origin , x-axis as real axis and y-axis as imaginary axis then,

x-coordinate = $\sqrt[m]{r} \cos (\frac{胃_{慰}}{m} + \frac{2 n 蟺}{m})$

y-coordinate= $\sqrt[m]{r} \sin (\frac{胃_{慰}}{m} + \frac{2 n 蟺}{m})$

magnitude of distance of the point from the origin is

$|z_{n}| = \sqrt{(\sqrt[m]{r} \cos {(\frac{胃_{慰}}{m} + \frac{2 n 蟺}{m}))}^{2} + (\sqrt[m]{r} \sin {(\frac{胃_{慰}}{m} + \frac{2 n 蟺}{m}))}^{2}} |z_{n}| = \sqrt{{(\sqrt[m]{r})}^{2} \{(\cos {(\frac{胃_{慰}}{m} + \frac{2 n 蟺}{m}))}^{2} + (\sin {(\frac{胃_{慰}}{m} + \frac{2 n 蟺}{m}))}^{2}\}} |z_{n}| = \sqrt[m]{r}$

Step 3. Conclusion

Now we can see that the for any value of n, magnitude of the distance of the point from the origin is constant. So the points form a circle with centre at the origin and radius equals to $\sqrt[m]{r}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Show that complex nth root of a nonzero complex number w lies on a circle with centre at the origin. What is the radius of this circle?

Short Answer

Step by step solution

Step 1. Considering a complex number w

Step 2. Calculations

Step 3. Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Statistics

Probability and Statistics

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.