Question

Which expression is a CUBE ROOT of -2i?

A. cubert(2) (cos(260 degree) + i sin(260 degree))
B. cubert(2) (cos(60 degree) + i sin(60 degree))
C. cubert(2) (cos(90 degree) + i sin(90 degree))
D. cubert(2) (cos(210 degree) + i sin(210 degree))

Answer 1

Would it be D? (Using improper math)

Answer 2

To approach this problem, you should write the complex number in polar form.

Answer 3

Ok! So it would be r = 2?

Answer 4

For simplicity, let's pull out the -2 for now.

Answer 5

Ok

Answer 6

Use Euler's identity to write -i.

Answer 7

I mean just i.

Answer 8

Would i = i?

Answer 9

Remember i is 90 degrees on the complex plane.

Answer 10

Oh, right!

Answer 11

So\[i = e^{ i \pi/2} \]

Answer 12

Now to find the cube root of just i, you can divide the exponent of the polar form by 3.

Answer 13

\[\sqrt[3]{i} = e^{i \pi/6}\]

Answer 14

Oh I see

Answer 15

Now rewrite the polar form into rectangular form.

Answer 16

And multiply with the cube root of -2, a negative real constant.

Answer 17

\[\sqrt[3]{2}= e ^{ipi/6}\]Sorry, I don't know how to do this area

Answer 18

Write it like this: \[e^{i \pi/6} = \cos(\pi/6) + i \sin(\pi/6)\]

Answer 19

Now in your question, it seems that they are representing the answers in degrees and also they are incorporating the negative factor into the complex part.

Answer 20

You will need to rewrite the angles in degree form, and also rotate by 90 degrees.

Answer 21

So \[\frac{ \pi }{ 6} = 30\]

Answer 22

Correct

Answer 23

And then it would be in the 3rd quadent?

Answer 24

Oh sorry, I meant rotate by 180 degrees.

Answer 25

Oh ok! So that makes it 210 degrees, which is answer D!

Answer 26

Correct

Answer 27

Great! Thank you for your help and taking the time to explain this to me!

Answer 28

It's my pleasure. Good luck.