Chapter 9: Q. 13 (page 631)

Find all the complex cube roots of $- 8 + 8 \sqrt{3} i$ . Then plot them in the complex plane.

Short Answer

Expert verified

The complex cube roots of $z_{0} = 2 \sqrt[3]{2} (\cos 40 ° + i \sin 40 °), z_{1} = 2 \sqrt[3]{2} (\cos 160 ° + i \sin 160 °), z_{2} = 2 \sqrt[3]{2} (\cos 280 ° + i \sin 280 °)$ .

Step by step solution

Step 1. Given Information

The given complex number is $- 8 + 8 \sqrt{3} i$

Complex root of a number in the form $w = r (\cos θ_{0} + i \sin θ_{0})$ is given by

$z_{k} = \sqrt[n]{r} [\cos (\frac{θ_{0}}{n} + \frac{2 k π}{n}) + i \sin (\frac{θ_{0}}{n} + \frac{2 k π}{n})]$

Step 2. Finding the value of r

$r = \sqrt{a^{2} + b^{2}} r = \sqrt{{(- 8)}^{2} + {(8 \sqrt{3})}^{2}} r = \sqrt{256} = 16$

Step 3. Finding the angle θ

$\tan θ = \frac{y}{x} = \frac{8 \sqrt{3}}{- 8} θ = \tan^{- 1} (\frac{8 \sqrt{3}}{- 8}) θ = - 60^{\circ}$

The given equation is in II quadrant, so $θ = 180 - 60 = 120^{\circ}$

Polar form of the given equation $- 8 + 8 \sqrt{3} = 16 (\cos 120^{\circ} + i \sin 120^{\circ})$

Step 4. Finding Complex Cube Roots

$z_{k} = \sqrt[3]{16} [\cos (\frac{120}{3} + \frac{360^{\circ} k}{3}) + i \sin (\frac{120}{3} + \frac{360^{\circ} k}{3})] z_{k} = 2 \sqrt[3]{2} [\cos (40^{\circ} + 120^{\circ} k) + i \sin (40^{\circ} + 120^{\circ} k)] w h e r e k = 0, 1, 2$

Step 5. Finding Different complex cube roots

At $k = 0,$ $z_{0} = 2 \sqrt[3]{2} [\cos 40^{\circ} + i \sin 40^{\circ}]$

At $k = 1,$ $z_{1} = 2 \sqrt[3]{2} [\cos (40^{\circ} + 120^{\circ}) + i \sin (40^{\circ} + 120^{\circ})] z_{1} = 2 \sqrt[3]{2} [\cos 160^{\circ} + i \sin 160^{\circ}]$

At $k = 2,$ $z_{2} = 2 \sqrt[3]{2} [\cos (40^{\circ} + 120^{\circ} \times 2) + i \sin (40^{\circ} + 120^{\circ} \times 2)] z_{2} = 2 \sqrt[3]{2} [\cos (280^{\circ}) + i \sin (280^{\circ})]$

Step 6. Graph of the Roots

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Find all the complex cube roots of $- 8 + 8 \sqrt{3} i$ . Then plot them in the complex plane.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the value of r

Step 3. Finding the angle θ

Step 4. Finding Complex Cube Roots

Step 5. Finding Different complex cube roots

Step 6. Graph of the Roots

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Find all the complex cube roots of-8+83i . Then plot them in the complex plane.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the value of r

Step 3. Finding the angle θ

Step 4. Finding Complex Cube Roots

Step 5. Finding Different complex cube roots

Step 6. Graph of the Roots

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Statistics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find all the complex cube roots of $- 8 + 8 \sqrt{3} i$ . Then plot them in the complex plane.