Chapter 9: Q. 55 (page 592)

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
55. The complex fourth roots of $4 - 4 \sqrt{3} i$ .

Short Answer

Expert verified

The complex fourth roots of $4 - 4 \sqrt{3} i$ are $\sqrt[4]{8} (\cos 75 + i \sin 75), \sqrt[4]{8} (\cos 165 + i \sin 165), \sqrt[4]{8} (\cos 255 + i \sin 255), \sqrt[4]{8} (\cos 345 + i \sin 345)$ .

Step by step solution

Step 1. Rewrite 4-43i in the polar form.

The magnitude of $4 - 4 \sqrt{3} i$ is

role="math" localid="1646848982162" $\sqrt{4^{2} + {(- 4 \sqrt{3})}^{2}} = \sqrt{16 + 48} = \sqrt{64} = 8$

which gives

$4 - 4 \sqrt{3} = 8 (\frac{1}{2} - \frac{\sqrt{3}}{2} i) = 8 (\cos 300 ° - i \sin 300 °)$

Step 2. Using the theorem of complex roots.

We get

$z_{k} = \sqrt[4]{8} [\cos (\frac{300}{4} + \frac{360 k}{4}) + i \sin (\frac{300}{4} + \frac{360 k}{4})]; k = 0, 1, 2, 3 = \sqrt[4]{8} [\cos (75 ° + 90 ° k) + i \sin (75 ° + 90 ° k)]$

Step 3. Substitute k=0,1,2,3 in zk=84[cos(75+90k)+isin(75+90k)].

We get

$z_{0} = \sqrt[4]{8} (\cos 75 + i \sin 75) z_{1} = \sqrt[4]{8} [\cos (75 + 90) + i \sin (75 + 90)] = \sqrt[4]{8} (\cos 165 + i \sin 165) z_{2} = \sqrt[4]{8} [\cos (75 + 180) + i \sin (75 + 180)] = \sqrt[4]{8} (\cos 255 + i \sin 255) z_{3} = \sqrt[4]{8} (\cos (75 + 270) + i \sin (75 + 270)) = \sqrt[4]{8} (\cos 345 + i \sin 345)$

So we have found the complex fourth roots of $4 - 4 \sqrt{3} i$ .

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 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
55. The complex fourth roots of $4 - 4 \sqrt{3} i$ .

Short Answer

Step by step solution

Step 1. Rewrite 4-43i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2,3 in zk=84[cos(75+90k)+isin(75+90k)].

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Probability and Statistics

Decision Maths

Logic and Functions

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.55. The complex fourth roots of4-43i.

Short Answer

Step by step solution

Step 1. Rewrite 4-43i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2,3 in zk=84[cos(75+90k)+isin(75+90k)].

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Probability and Statistics

Decision Maths

Logic and Functions

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
55. The complex fourth roots of $4 - 4 \sqrt{3} i$ .