anonymous one year ago 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))

1. anonymous

Would it be D? (Using improper math)

2. anonymous

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

3. anonymous

Ok! So it would be r = 2?

4. anonymous

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

5. anonymous

Ok

6. anonymous

Use Euler's identity to write -i.

7. anonymous

I mean just i.

8. anonymous

Would i = i?

9. anonymous

Remember i is 90 degrees on the complex plane.

10. anonymous

Oh, right!

11. anonymous

So$i = e^{ i \pi/2}$

12. anonymous

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

13. anonymous

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

14. anonymous

Oh I see

15. anonymous

Now rewrite the polar form into rectangular form.

16. anonymous

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

17. anonymous

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

18. anonymous

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

19. anonymous

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.

20. anonymous

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

21. anonymous

So $\frac{ \pi }{ 6} = 30$

22. anonymous

Correct

23. anonymous

And then it would be in the 3rd quadent?

24. anonymous

Oh sorry, I meant rotate by 180 degrees.

25. anonymous

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

26. anonymous

Correct

27. anonymous

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

28. anonymous

It's my pleasure. Good luck.