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Math Help - Find all complex numbers z

  1. #1
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    Find all complex numbers z

    Find all complex numbers z so that z3 = -8i.

    I know I have to do something with polar form, but I'm really lost.
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  2. #2
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    Re: Find all complex numbers z

    If z=re^{i\varphi}, then z^3=r^3e^{3i\varphi}. Note that -i=e^{i(3\pi/2)}. So you need to find all 0\le\varphi<2\pi such that 3\varphi\equiv3\pi/2\pmod{2\pi}. There are three such \varphi. And, of course, you need to find r.
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  3. #3
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    Re: Find all complex numbers z

    Quote Originally Posted by biggerleaffan View Post
    Find all complex numbers z so that z3 = -8i.
    I know I have to do something with polar form, but I'm really lost.
    This is posted in basic algebra forum. Reply #2 is correct but advanced.

    Notation: r\exp(i\theta)=r\cos(\theta)+i~r\sin(\theta)~. So {\left[ {r\exp \left( {i\theta } \right)} \right]^n} = {r^n}\exp \left( {i\theta n} \right)

    Let \rho  = 2\exp \left( {\frac{{ - i\pi }}{6}} \right), so \rho^3=-8i. We have one root.

    Let \xi  = \exp \left( {\frac{{2\pi i}}{3}} \right), then you need to show that \rho\cdot\xi~\&~\rho\cdot\xi^2 are the other two roots.
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