Chapter 9: Q. 57 (page 592)

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
57. The complex fourth roots of $- 16 i$ .

Short Answer

Expert verified

The complex fourth roots of $- 16 i$ are $2 (\cos 67.5 + i \sin 67.5), 2 (\cos 157.5 + i \sin 157.5), 2 (\cos 247.5 + i \sin 247.5), 2 (\cos 337.5 + i \sin 337.5)$ .

Step by step solution

Step 1. Rewrite -16i in the polar form.

The magnitude of $- 16 i$ is

$\sqrt{0^{2} + {(- 16)}^{2}} = \sqrt{256} = 16$

which gives

$- 16 i = 16 (0 - i) = 16 (\cos 270 ° - i \sin 270 °)$

Step 2. Using the theorem of complex roots.

We get

$z_{k} = \sqrt[4]{16} [\cos (\frac{270}{4} + \frac{360 k}{4}) + i \sin (\frac{270}{4} + \frac{360 k}{4})]; k = 0, 1, 2, 3 = 2 [\cos (67.5 ° + 90 ° k) + i \sin (67.5 ° + 90 ° k)]$

Step 3. Substitute k=0,1,2,3 in zk=2[cos(67.5+90k)+isin(67.5+90k)].

We get

$z_{0} = 2 (\cos 37.5 + i \sin 67.5) z_{1} = 2 [\cos (67.5 + 90) + i \sin (67.5 + 90)] = 2 (\cos 157.5 + i \sin 157.5) z_{2} = 2 [\cos (67.5 + 180) + i \sin (67.5 + 180)] = 2 (\cos 247.5 + i \sin 247.5) z_{3} = 2 [\cos (67.5 + 270) + i \sin (67.5 + 270)] = 2 (\cos 337.5 + i \sin 337.5)$

So we have found the complex fourth roots of $- 16 i$ .

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In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
57. The complex fourth roots of $- 16 i$ .

Short Answer

Step by step solution

Step 1. Rewrite -16i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2,3 in zk=2[cos(67.5+90k)+isin(67.5+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.57. The complex fourth roots of-16i.

Short Answer

Step by step solution

Step 1. Rewrite -16i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2,3 in zk=2[cos(67.5+90k)+isin(67.5+90k)].

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Logic and Functions

Calculus

Probability and Statistics

Pure 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.
57. The complex fourth roots of $- 16 i$ .