Chapter 11: Problem 2

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

Short Answer

Expert verified

Answer: No, we cannot deduce that \((2+2\mathrm{j})^8 = 4096\), because we found that \(z^8 = 32768\), not \(4096\).

Step by step solution

Find the magnitude r of the complex number

To find the magnitude \(r\) of the complex number \(z = 2 + 2\mathrm{j}\), we use the formula \(r = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts of \(z\), respectively. In this case, \(a = 2\) and \(b = 2\). So, \(r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}\).

Find the argument θ of the complex number

To find the argument \(\theta\) of the complex number \(z = 2 + 2\mathrm{j}\), we use the formula \(\theta = \arctan(\frac{b}{a})\), where \(a\) and \(b\) are the real and imaginary parts of \(z\), respectively. In this case, \(a = 2\) and \(b = 2\). So, \(\theta = \arctan(\frac{2}{2}) = \arctan(1) = \frac{\pi}{4}\).

Convert the complex number to polar form

Now that we have found the magnitude and argument of \(z\), we can express it in polar form. The polar form of \(z\) is given by \(z = r(\cos\theta + \mathrm{j}\sin\theta)\), where \(r = 2\sqrt{2}\) and \(\theta = \frac{\pi}{4}\). So, \(z = 2\sqrt{2}\left(\cos\frac{\pi}{4} + \mathrm{j}\sin\frac{\pi}{4}\right)\).

Apply De Moivre's Theorem to find z⁸

Now, we apply De Moivre's theorem to find \(z^8\). De Moivre's theorem states that \((r(\cos\theta + \mathrm{j}\sin\theta))^n = r^n(\cos(n\theta) + \mathrm{j}\sin(n\theta))\). In this case, \(n = 8\). So, \(z^8 = (2\sqrt{2})^8\left(\cos(8\cdot\frac{\pi}{4}) + \mathrm{j}\sin(8\cdot\frac{\pi}{4})\right)\).

Calculate z⁸

Now, let's calculate \(z^8\). First, we find the magnitude and argument of \(z^8\). \((2\sqrt{2})^8 = (2^8)(\sqrt{2}^8) = 256\cdot128 = 32768\). The argument of \(z^8\) is \(8\cdot\frac{\pi}{4} = 2\pi\). Now, since \(2\pi\) is a multiple of \(2\pi\), we have \(\cos(2\pi) = 1\) and \(\sin(2\pi) = 0\). So, \(z^8 = 32768\left(\cos(2\pi) + \mathrm{j}\sin(2\pi)\right) = 32768(1 + 0\mathrm{j}) = 32768\).

Compare z⁸ with the given result

We found that \(z^8 = 32768\). However, the problem asks us to deduce that \((2+2\mathrm{j})^8 = 4096\). Since our calculated value of \(z^8\) is not equal to \(4096\), we cannot deduce that \((2+2\mathrm{j})^8 = 4096\). There may be a mistake in the given value of \((2+2\mathrm{j})^8\).

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!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Find the magnitude r of the complex number

Find the argument θ of the complex number

Convert the complex number to polar form

Apply De Moivre's Theorem to find z⁸

Calculate z⁸

Compare z⁸ with the given result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Wave Optics

Dynamics

Atoms and Radioactivity

Circular Motion and Gravitation

Force

Study anywhere. Anytime. Across all devices.