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Complex powers

  1. Apr 29, 2015 #1
    Calculate

    ( minus ( 2 over 3 ) + ( 2 over 3 ) i ) to the power minus 4,

    simplifing your answer and giving it in the form a + i b, with a and b given exactly.


    I found the modulus by:
    sqrt((-2/3)^2 + (2/3)^2)
    = (2*sqrt(2))/3

    the argument is:
    pi - 1 (from a sketch in the complex plane)

    hence:
    -2/3 + 2/3i = (2*sqrt(2)/3)*(cos(pi-1)+isin(pi-1))

    using de moivres formula:
    (-2/3 + 2/3i)^4 = (2*sqrt(2)/3)^4*(cos(4(pi-1))+isin(4(pi-1)))

    but what next? I know I need to convert to cartesian form but how?
     
  2. jcsd
  3. Apr 29, 2015 #2

    Ray Vickson

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    How do you write ##r e^{i \theta} = r \cos(\theta) + i r \sin(\theta)## in the form ##x + iy##?
     
  4. Apr 29, 2015 #3

    SteamKing

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    This looks OK.
    How did you determine this? Show your sketch, if necessary.

    Here is a nifty graphic:

    alg2-nb35-15.jpg
     
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