Question

Find the 4th roots 2^3+2i

Answer 1

\begin{eqnarray*}z^4 = r^4e^{4i\theta} &=& 2^3 + 2i \\ &=& 2(4 + i) \\ &=& 2 \sqrt{4^2 + 1^2}\exp(i \arctan(1/4)) \\&=&2\sqrt{17} \exp(i \arctan(1/4))\end{eqnarray*}therefore,\begin{eqnarray*}&&r = \sqrt[4]{2\sqrt{17}} \land 4\theta = \arctan(1/4) + 2k\pi ,\quad k = 0,1,2,3\\\Rightarrow&&r = \sqrt[4]{2\sqrt{17}}\land \theta = \frac{\arctan(1/4)+2k\pi}{4} ,\quad k = 0,1,2,3.\end{eqnarray*}