Problem
Find the cube root of \[ 4-4 \sqrt{3} i \] that graphs in the second quadrant. \[ [?]\left(\cos []^{\circ}+i \sin []^{\circ}\right) \]
Solution
Step 1 ：\[r = \sqrt[6]{\left|4-4\sqrt{3}i\right|} = \sqrt[6]{64}\]
Step 2 ：\[\theta = \left(\frac{\arg(4-4\sqrt{3}i)}{3}\right) = \left(\frac{\arctan\left(\frac{-4\sqrt{3}}{4}\right)}{3}\right) = \frac{300^\circ}{3}\]
Step 3 ：\[\text{Cube root} = \left(r\cos \left(\frac{\theta}{3}\right) + ir\sin \left(\frac{\theta}{3}\right)\right) = 2\left(\cos 100^\circ + i\sin 100^\circ\right)\]
