anonymous
  • anonymous
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))
Mathematics
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anonymous
  • anonymous
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))
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
Would it be D? (Using improper math)
anonymous
  • anonymous
To approach this problem, you should write the complex number in polar form.
anonymous
  • anonymous
Ok! So it would be r = 2?

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anonymous
  • anonymous
For simplicity, let's pull out the -2 for now.
anonymous
  • anonymous
Ok
anonymous
  • anonymous
Use Euler's identity to write -i.
anonymous
  • anonymous
I mean just i.
anonymous
  • anonymous
Would i = i?
anonymous
  • anonymous
Remember i is 90 degrees on the complex plane.
anonymous
  • anonymous
Oh, right!
anonymous
  • anonymous
So\[i = e^{ i \pi/2} \]
anonymous
  • anonymous
Now to find the cube root of just i, you can divide the exponent of the polar form by 3.
anonymous
  • anonymous
\[\sqrt[3]{i} = e^{i \pi/6}\]
anonymous
  • anonymous
Oh I see
anonymous
  • anonymous
Now rewrite the polar form into rectangular form.
anonymous
  • anonymous
And multiply with the cube root of -2, a negative real constant.
anonymous
  • anonymous
\[\sqrt[3]{2}= e ^{ipi/6}\]Sorry, I don't know how to do this area
anonymous
  • anonymous
Write it like this: \[e^{i \pi/6} = \cos(\pi/6) + i \sin(\pi/6)\]
anonymous
  • 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.
anonymous
  • anonymous
You will need to rewrite the angles in degree form, and also rotate by 90 degrees.
anonymous
  • anonymous
So \[\frac{ \pi }{ 6} = 30\]
anonymous
  • anonymous
Correct
anonymous
  • anonymous
And then it would be in the 3rd quadent?
anonymous
  • anonymous
Oh sorry, I meant rotate by 180 degrees.
anonymous
  • anonymous
Oh ok! So that makes it 210 degrees, which is answer D!
anonymous
  • anonymous
Correct
anonymous
  • anonymous
Great! Thank you for your help and taking the time to explain this to me!
anonymous
  • anonymous
It's my pleasure. Good luck.

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