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Math Help - How do I find the cube roots of this?

  1. #1
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    How do I find the cube roots of this?

    I had this question on a test I recently took and I am still totally unaware of how to solve it. If anyone could help I would greatly appreciate it. Thanks!

    Find the cube roots of 3sqrt(3) - 3i
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  2. #2
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    Re: How do I find the cube roots of this?

    \displaystyle \begin{align*} 3\sqrt{3} - 3i &= 3\left( \sqrt{3} - i \right) \\ &= 3 \left( 2 \, e^{ -\frac{\pi}{6} \, i } \right) \\ &= 6 \, e^{-\frac{\pi}{6} \, i } \end{align*}

    So the cube roots can be found by taking this to the power of \displaystyle \begin{align*} \frac{1}{3} \end{align*} and remembering that there will be three roots evenly spaced around the circle, so each separated by an angle of \displaystyle \begin{align*} \frac{2\pi}{3} \end{align*}.

    The first cube root is

    \displaystyle \begin{align*} \left( 6\, e^{-\frac{\pi}{6}\, i }\right) ^{\frac{1}{3}} &= \sqrt[3]{6} \, e^{ -\frac{\pi}{18} \, i  } \end{align*}

    and the other two cube roots are \displaystyle \begin{align*} \sqrt[3]{6}\, e^{\left( -\frac{\pi}{18} - \frac{2\pi}{3} \right) i} = \sqrt[3]{6}\, e^{ -\frac{13\pi}{18} \, i } \end{align*} and \displaystyle \begin{align*} \sqrt[3]{6}\, e^{\left( -\frac{\pi}{18} + \frac{2\pi}{3} \right) i} = \sqrt[3]{6}\, e^{\frac{11\pi}{18} \, i } \end{align*}
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    Re: How do I find the cube roots of this?

    I can't believe I forgot about e! Thank you very much, Prove it.
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  4. #4
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    Re: How do I find the cube roots of this?

    Just something to note, since you were given the original complex number in Cartesians, you should really give your answers in Cartesians as well.
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