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