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Math Help - roots of complex numbers

  1. #1
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    roots of complex numbers

    a) find all roots of the equation z^5=sqrt(3)*i-1, present your answer in polar form

    b) express (2-2i)^5 in exact polar form
    (-1+sqrt(3)*i)^4

    I was alright until the whole polar form thing, your help would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by samdmansam View Post
    a) find all roots of the equation z^5=sqrt(3)*i-1, present your answer in polar form
    For a) what is the polar form of -1 + i\sqrt{3}?

    r~cos(\theta) + ir~sin(\theta) = -1 + i\sqrt{3}

    So
    r~cos(\theta) = -1
    and
    r~sin(\theta) = \sqrt{3}

    Thus
    \frac{r~sin(\theta)}{r~cos(\theta)} = -\sqrt{3}

    Now,
    tan(\theta) = \left | -\sqrt{3} \right | \implies \theta = \frac{\pi}{3}

    and since sine is positive and cosine negative we know the reference angle is in QII. Thus
    \theta = \frac{2 \pi}{3}

    Then
    r~cos(\theta) = -1

    r \cdot -\frac{1}{2} = -1

    r = 2

    So
    z^5 = 2e^{2i \pi / 3}

    Can you solve it from here? (Please note that there are five answers to this.)

    -Dan
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