Chapter 9: Q. 54 (page 592)

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

Short Answer

Expert verified

The complex four roots of $\sqrt{3} - i$ are $\sqrt[4]{2} (\cos 82.5 + i \sin 82.5), \sqrt[4]{2} (\cos 172.5 + i \sin 172.5), \sqrt[4]{2} (\cos 262.5 + i \sin 262.5), \sqrt[4]{2} (\cos 352.5 + i \sin 352.5)$ .

Step by step solution

Step 1. Rewrite z=3-i in the polar form.

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

$\sqrt{{(\sqrt{3})}^{2} + {(- 1)}^{2}} = \sqrt{3 + 1} = \sqrt{4} = 2$

which gives

$\sqrt{3} - i = 2 (\frac{\sqrt{3}}{2} - i \frac{1}{2}) = 2 (\cos 330 ° + i \sin 330 °)$

Step 2. Using De Moivre's Theorem.

We get

${(\sqrt{3} - i)}^{\frac{1}{4}} = {[2 (\cos 330 ° + i \sin 330 °)]}^{\frac{1}{4}} z_{k} = {(2)^{\frac{1}{4}} [\cos (\frac{330}{4} + \frac{360 k}{4}) + i \sin (\frac{330}{4} + \frac{360 k}{4})]}; k = 0, 1, 2, 3 = \sqrt[4]{2} [\cos (82.5 + 90 k) + i \sin (82.5 + 90 k)]; k = 0, 1, 2, 3$

Step 3. Substitute k=0,1,2,3 in zk=24[cos(82.5+90k)+isin(82.5+90k)].

We get

$z_{0} = \sqrt[4]{2} (\cos 82.5 + i \sin 82.5) z_{1} = \sqrt[4]{2} (\cos (82.5 + 90) + \sin (82.5 + 90)) = \sqrt[4]{2} (\cos 172.5 + i \sin 172.5) z_{2} = \sqrt[4]{2} [\cos (82.5 + 180) + i \sin (82.5 + 180)] = \sqrt[4]{2} (\cos 262.5 + i \sin 262.5) z_{3} = \sqrt[4]{2} [\cos (82.5 + 270) + i \sin (82.5 + 270)] = \sqrt[4]{2} (\cos 352.5 + i \sin 352.5)$

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

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

Short Answer

Step by step solution

Step 1. Rewrite z=3-i in the polar form.

Step 2. Using De Moivre's Theorem.

Step 3. Substitute k=0,1,2,3 in zk=24[cos(82.5+90k)+isin(82.5+90k)].

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In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.54. The complex fourth roots of3-i.

Short Answer

Step by step solution

Step 1. Rewrite z=3-i in the polar form.

Step 2. Using De Moivre's Theorem.

Step 3. Substitute k=0,1,2,3 in zk=24[cos(82.5+90k)+isin(82.5+90k)].

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Mechanics Maths

Geometry

Applied Mathematics

Discrete Mathematics

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.
54. The complex fourth roots of $\sqrt{3} - i$ .