Which of the following is not a root of z^3 = 1 - sqrt[3] to the nearest hundredth?

Question

I have four options:

 

A.) -0.22 + 1.24i

B.) -0.97 - 0.81i

C.) 1.02 - 0.65i

D.) 1.18 - 0.43i

 

I really need help figuring out how to attack this problem. I have already converted it to polar form and got:

 

2(cos(5pi/3 + 2npi) + i sin(5pi/3 + 2npi))

 

After that step, I'm lost. Please help, I want to know how to work these types of problems out.

Answer 1

There are three cube-roots of 1 - i√3 = 2 cis 5π/3 are 2^1/3 cis (5/9 + 2k/3)π = 2^1/3 cis (6k+5)π/9:

 

2^1/3 cis 5π/9 ≈ -0.22 + 1.24i   (This is choice A)

 

2^1/3 cis 11π/9 ≈ -0.97 - 0.81i  (This is choice B)

 

2^1/3 cis 17π/9 ≈ 1.18 - 0.34i   (This is choice D)  I think you typed that one.

 

So the answer is choice C.