standard form

Jun 2008
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Illinois
Use De Moivre's theorem: \(\displaystyle (r \cdot cis\theta )^n = r^n \cdot cis(n \theta) \), where \(\displaystyle cis(x) \) is defined as: \(\displaystyle cis(x) = \cos(x) + i \sin(x) \).

Here \(\displaystyle (1+i) = \sqrt 2 cis(\frac {\pi} 4)\) so:

\(\displaystyle
(1+i)^{20} = (\sqrt 2)^{20} \cdot cis (20 \times \frac {\pi} 4 )= 2^{10} cis(5\pi) = 2^{10}(\cos(5\pi) + i \sin(5\pi)) = 2^{10}
\)
 
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