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Math Help - complex number exponentials

  1. #1
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    complex number exponentials

    simplify, in cartesian and polar forms:

    (sqrt(3)-3i)^10

    (sqrt(3)-3i)^ -1

    (sqrt(3)-3i)^ 1/3


    can someone please explain how i would go about doing this?
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  2. #2
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    You must learn to find the arguments of complex numbers!
    \arg \left( {\sqrt 3  - 3i} \right) = \arctan \left( {\frac{{ - 3}}{{\sqrt 3 }}} \right) = \arctan \left( { - \sqrt 3 } \right) =  - \frac{\pi }{3}.

    Now all the three questions depend upon two facts: \arg \left( {\sqrt 3  - 3i} \right) =  - \frac{\pi }{3} and \left| {\sqrt 3  - 3i} \right| = \sqrt {12}  = 2\sqrt{3}
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  3. #3
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    okay, so now what?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by mistykz View Post
    okay, so now what?
    we can express the complex number x + iy as r e^{i \theta}, where r = |x + iy| = \sqrt {x^2 + y^2} and \theta = \arg (x + iy)

    in this form, it is easy to apply the powers and simplify

    you can see the "Background" section in the first post here for more information.
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