Find all values of \((1+i)^{1/3}\).

Polar form: \(1+i=\sqrt2\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)=\sqrt2\left[\cos\left(2k\pi+\frac{\pi}{4}\right)+i\sin\left(2k\pi+\frac{\pi}{4}\right)\right]\).

De Moivre: \((1+i)^{1/3}=2^{1/6}\left[\cos\frac{(8k+1)\pi}{12}+i\sin\frac{(8k+1)\pi}{12}\right],\ k=0,1,2\).

\(k=0:\ 2^{1/6}\left(\cos\frac{\pi}{12}+i\sin\frac{\pi}{12}\right)=1.0842+0.2905i\)
\(k=1:\ 2^{1/6}\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right)=-0.7937+0.7937i\)
\(k=2:\ 2^{1/6}\left(\cos\frac{17\pi}{12}+i\sin\frac{17\pi}{12}\right)=-0.2905-1.0842i\)

(Check: the three cube roots add to zero.) The three values are \(2^{1/6}\,\mathrm{cis}\frac{\pi}{12},\ 2^{1/6}\,\mathrm{cis}\frac{3\pi}{4},\ 2^{1/6}\,\mathrm{cis}\frac{17\pi}{12}\).