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Math Help - Euler's form

  1. #1
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    Euler's form

    Hi,

    steps appreciated in the question:

    [squareroot(2) - i*squareroot(2)]^10

    thanks
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Moon Hoplite View Post
    Hi,

    steps appreciated in the question:

    [squareroot(2) - i*squareroot(2)]^10

    thanks
    You need to apply De Moivre's Theorem

    First, covert this to polar form:

    You should get z=2e^{i\left(\frac{3\pi}{4}\right)}

    Thus, z^{10}=2^{10}e^{i\left(\frac{30\pi}{4}\right)}=2^{  10}e^{i\left(-\frac{\pi}{2}+8\pi\right)}=2^{10}\left[\cos\!\left(-\frac{\pi}{2}\right)+i\sin\!\left(-\frac{\pi}{2}\right)\right]=-2^{10}i=\color{red}\boxed{-1024i}

    Does this make sense?
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