Euler's form

**Moon Hoplite** · February 2nd 2009, 02:17 PM

Hi,

steps appreciated in the question:

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

thanks

**Chris L T521** · February 2nd 2009, 02:26 PM

Originally Posted by Moon Hoplite

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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