Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Problem 1

If \(z=4 \angle \frac{\pi}{6}\) find \(z^{6}\) in polar form.

Short Answer

Expert verified
Answer: The sixth power of the given complex number in polar form is \(z^6 = 4096\angle \pi\).

Step by step solution

01

Identify the given complex number in polar form

The given complex number is \(z=4\angle\frac{\pi}{6}\). Here, \(r=4\) and \(\theta=\frac{\pi}{6}\).
02

Apply De Moivre's theorem

According to De Moivre's theorem, \((r\angle\theta)^n = r^n\angle n\theta\). In this case, we need to find \(z^6\), so we have \(n=6\). Plug in the values of \(r\), \(\theta\), and \(n\) to find the sixth power of \(z\): \((4\angle\frac{\pi}{6})^6 = 4^6\angle 6\frac{\pi}{6}\)
03

Calculate the result

Using the values we found in the previous step, calculate the result for \(z^6\): \(4^6\angle 6\frac{\pi}{6} = 4096\angle \pi\)
04

Write the final answer in polar form

So, the sixth power of the given complex number in polar form is: \(z^6 = 4096\angle \pi\)

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Solve the equation \(-z^{2}+3 z-4=0\)

Express each of the following in terms of

\(2 \quad\) Express \(z=2+2 \mathrm{j}\) in polar form and hence find \(z^{8}\), leaving your answer in polar form. Deduce that \((2+2 \mathrm{j})^{8}=4096\)

A capacitor and resistor are placed in parallel. Show that the complex impedance of this combination is given by $$ \frac{1}{Z}=\frac{1}{R}+j \omega C $$ Find an expression for \(Z\).

Show that \(3\left(\frac{\mathrm{e}^{j \omega}-\mathrm{e}^{-j \omega}}{j \omega}\right)=\frac{6 \sin \omega}{\omega}\)
See all solutions

Recommended explanations on Physics Textbooks

Energy Physics

Read Explanation

Rotational Dynamics

Read Explanation

Force

Read Explanation

Wave Optics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Linear Momentum

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free