# standard form

#### kenzie103109

please write (1 + i )^20 in standard form

#### ebaines

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:

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

kenzie103109