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

Answer

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Answer

\[\text{Cube root} = \left(r\cos \left(\frac{\theta}{3}\right) + ir\sin \left(\frac{\theta}{3}\right)\right) = 2\left(\cos 100^\circ + i\sin 100^\circ\right)\]

Steps

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 ：\[\text{Cube root} = \left(r\cos \left(\frac{\theta}{3}\right) + ir\sin \left(\frac{\theta}{3}\right)\right) = 2\left(\cos 100^\circ + i\sin 100^\circ\right)\]

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

Answer

Popular Problems

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