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Chapter 9: Q. 58 (page 592)

In Problems 53鈥�60, find all the complex roots. Leave your answers in polar form with the argument in degrees.

58. The complex cube roots of-8.

Short Answer

Expert verified

The complex cube roots of -8are2(cos60+isin60),2(cos180+isin180),2(cos300+isin300).

Step by step solution

01

Step 1. Rewrite -8 in the polar form.

The magnitude of -8is

(-8)2+0=8

which gives

-8=8(-1-0i)=8(cos180掳+isin180掳)

02

Step 2. Using the Theorem of complex roots.

We get

zk=83[cos(1803+360k3)+isin(1803+360k3);k=0,1,2=2[cos(60+120k)+isin(60+120k)]

03

Step 3. Substitute k=0,1,2 in zk=2(cos(60+120k)+isin(60+120k)).

We get

z0=2(cos60+isin60)z1=2(cos(60+120)+isin(60+120))=2(cos180+isin180)z2=2(cos(60+240)+isin(60+240))=2(cos300+isin300)

So we have found the complex cube roots of-8.

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