Find the cube root of
$4 - 4 \sqrt{3} i$
that graphs in the second quadrant.
$[?] (\cos []^{\circ} + i \sin []^{\circ})$

Answer

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Answer

$Cube root = (r \cos (\frac{θ}{3}) + i r \sin (\frac{θ}{3})) = 2 (\cos 100^{\circ} + i \sin 100^{\circ})$

Steps

Step 1 ： $r = \sqrt[6]{| 4 - 4 \sqrt{3} i |} = \sqrt[6]{64}$

Step 2 ： $θ = (\frac{\arg (4 - 4 \sqrt{3} i)}{3}) = (\frac{\arctan (\frac{- 4 \sqrt{3}}{4})}{3}) = \frac{300^{\circ}}{3}$

Step 3 ： $Cube root = (r \cos (\frac{θ}{3}) + i r \sin (\frac{θ}{3})) = 2 (\cos 100^{\circ} + i \sin 100^{\circ})$