Multiplying $a + bi$ by $c + di$ in rectangular form is a chore. In polar form it becomes almost trivial, and powers become easy. The reason is the sum formulas.
Key idea
$$r_1(\cos\alpha + i\sin\alpha) \cdot r_2(\cos\beta + i\sin\beta) = r_1r_2\big(\cos(\alpha + \beta) + i\sin(\alpha + \beta)\big)$$ Multiply the moduli and add the angles; to divide, divide the moduli and subtract the angles. Repeating gives De Moivre's theorem: $$\big[r(\cos\theta + i\sin\theta)\big]^n = r^n(\cos n\theta + i\sin n\theta).$$
Why it works
Multiply out: $$(\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta) = (\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\sin\alpha\cos\beta + \cos\alpha\sin\beta),$$ using $i^2 = -1$. The real part is $\cos(\alpha + \beta)$ and the imaginary part is $\sin(\alpha + \beta)$. So multiplying by a complex number stretches by its modulus and rotates by its angle. Doing that $n$ times rotates $n$ times.
Worked examples
Worked example: A product
Multiply $2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)$ by $3\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$.
- Moduli: $2 \cdot 3 = 6$. Angles: $\frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2}$.
- $6\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right) = 6(0 + i)$.
Answer: $6i$
Worked example: A big power
Find $(1 + i)^8$.
- $1 + i = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)$.
- De Moivre: $\left(\sqrt{2}\right)^8 = 16$ and $8 \cdot \frac{\pi}{4} = 2\pi$.
- $16(\cos 2\pi + i\sin 2\pi) = 16$.
Answer: $16$
Worked example: Back to rectangular
Find $\left(1 + i\sqrt{3}\right)^5$.
- $1 + i\sqrt{3} = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$.
- $2^5 = 32$ and $5 \cdot \frac{\pi}{3} = \frac{5\pi}{3}$.
- $32\left(\frac{1}{2} - \frac{\sqrt{3}}{2}i\right) = 16 - 16\sqrt{3}\,i$.
Answer: $16 - 16\sqrt{3}\,i$
Common mistake
Multiplying the angles instead of adding them, as in $\frac{\pi}{6} \cdot \frac{\pi}{3}$. Angles add because rotations combine: turn $30^\circ$, then $60^\circ$, and you have turned $90^\circ$. Only in a power, $z^n$, does the angle get multiplied, by the whole number $n$, because you add it $n$ times.