Chapter 9: Q. 49 (page 592)

In Problems 41–52, write each expression in the standard form $a + b i$ .
49. ${(1 - i)}^{5}$

Short Answer

Expert verified

${(1 - i)}^{5} = - 4 + 4 i$ .

Step by step solution

Step 1. Rewrite 1-i in the form r(cos∅+isin∅).

The magnitude of $1 - i$ is

$\sqrt{1^{2} + {(- 1)}^{2}} = \sqrt{1 + 1} = \sqrt{2}$

which gives

$(1 - i) = \sqrt{2} (\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} i) = \sqrt{2} (\cos \frac{π}{4} - i \sin \frac{π}{4})$

Step 2. Using De Moivre's Theorem.

We know that if a complex number is $z = r (\cos \emptyset + i \sin \emptyset)$ then $z^{n} = r^{n} [\cos (n \emptyset) + i \sin (n \emptyset)]$ .

This gives

role="math" localid="1646763868779" ${(1 - i)}^{5} = {[\sqrt{2} (\cos \frac{π}{4} - i \sin \frac{π}{4})]}^{5} = [(\sqrt{2})^{5} (\cos {5 \times \frac{π}{4}} - i \sin {5 \times \frac{π}{4}})] = 4 \sqrt{2} (\cos \frac{5 π}{4} - i \sin \frac{5 π}{4}) = 4 \sqrt{2} (- \frac{1}{\sqrt{2}} - i (- \frac{1}{\sqrt{2}})) = - 4 + 4 i$

So, ${(1 - i)}^{5} = - 4 + 4 i$

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 Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems 41–52, write each expression in the standard form $a + b i$ .
49. ${(1 - i)}^{5}$

Short Answer

Step by step solution

Step 1. Rewrite 1-i in the form r(cos∅+isin∅).

Step 2. Using De Moivre's Theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Discrete Mathematics

Mechanics Maths

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 41–52, write each expression in the standard form a+bi.49.(1-i)5

Short Answer

Step by step solution

Step 1. Rewrite 1-i in the form r(cos∅+isin∅).

Step 2. Using De Moivre's Theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Discrete Mathematics

Mechanics Maths

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 41–52, write each expression in the standard form $a + b i$ .
49. ${(1 - i)}^{5}$