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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))

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  1. anonymous
    • one year ago
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    Would it be D? (Using improper math)

  2. anonymous
    • one year ago
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    To approach this problem, you should write the complex number in polar form.

  3. anonymous
    • one year ago
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    Ok! So it would be r = 2?

  4. anonymous
    • one year ago
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    For simplicity, let's pull out the -2 for now.

  5. anonymous
    • one year ago
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    Ok

  6. anonymous
    • one year ago
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    Use Euler's identity to write -i.

  7. anonymous
    • one year ago
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    I mean just i.

  8. anonymous
    • one year ago
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    Would i = i?

  9. anonymous
    • one year ago
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    Remember i is 90 degrees on the complex plane.

  10. anonymous
    • one year ago
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    Oh, right!

  11. anonymous
    • one year ago
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    So\[i = e^{ i \pi/2} \]

  12. anonymous
    • one year ago
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    Now to find the cube root of just i, you can divide the exponent of the polar form by 3.

  13. anonymous
    • one year ago
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    \[\sqrt[3]{i} = e^{i \pi/6}\]

  14. anonymous
    • one year ago
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    Oh I see

  15. anonymous
    • one year ago
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    Now rewrite the polar form into rectangular form.

  16. anonymous
    • one year ago
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    And multiply with the cube root of -2, a negative real constant.

  17. anonymous
    • one year ago
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    \[\sqrt[3]{2}= e ^{ipi/6}\]Sorry, I don't know how to do this area

  18. anonymous
    • one year ago
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    Write it like this: \[e^{i \pi/6} = \cos(\pi/6) + i \sin(\pi/6)\]

  19. anonymous
    • one year ago
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    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
    • one year ago
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    You will need to rewrite the angles in degree form, and also rotate by 90 degrees.

  21. anonymous
    • one year ago
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    So \[\frac{ \pi }{ 6} = 30\]

  22. anonymous
    • one year ago
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    Correct

  23. anonymous
    • one year ago
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    And then it would be in the 3rd quadent?

  24. anonymous
    • one year ago
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    Oh sorry, I meant rotate by 180 degrees.

  25. anonymous
    • one year ago
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    Oh ok! So that makes it 210 degrees, which is answer D!

  26. anonymous
    • one year ago
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    Correct

  27. anonymous
    • one year ago
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    Great! Thank you for your help and taking the time to explain this to me!

  28. anonymous
    • one year ago
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    It's my pleasure. Good luck.

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