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# How do I use DeMoivre's theorem to find (1-i)^10?

Nov 4, 2015

#### Answer:

$- 32 i$

#### Explanation:

First write this complex number in polar form and then apply De Moivre :

${\left(1 - i\right)}^{10} = {\left(\sqrt{2} \angle - \frac{\pi}{4}\right)}^{10} = {\left[\sqrt{2} \left(\cos \left(- \frac{\pi}{4}\right) + i \sin \left(- \frac{\pi}{4}\right)\right)\right]}^{10}$

$= {\left(\sqrt{2}\right)}^{10} \left[\cos \left(- 10 \frac{\pi}{4}\right) + i \sin \left(- 10 \frac{\pi}{4}\right)\right]$

$= - 32 i$