Find the fifth root of
\[
32 i
\]
that graphs in the first quadrant.
\[
2\left(\cos [?]^{\circ}+i \sin [?]^{\circ}\right)
\]
Enter your answer in degrees.

Answer

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Answer

Determine the graph in the first quadrant: \(k = 0, 2\left(\cos 18^\circ + i \sin 18^\circ\right)\)

Steps

Step 1 ：Convert 32i to polar form: \(32i = 32(\cos 90^\circ + i\sin 90^\circ)\)

Step 2 ：Apply 5th root: \(\sqrt[5]{32}\left(\cos\frac{90^\circ + 360^\circ k}{5} + i \sin\frac{90^\circ + 360^\circ k}{5}\right), k = 0, 1, 2, 3, 4\)

Step 3 ：Determine the graph in the first quadrant: \(k = 0, 2\left(\cos 18^\circ + i \sin 18^\circ\right)\)