Roots of Complex Number/Examples/6th Roots of -27 i

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Example of Roots of Complex Number

The complex $6$th roots of $-27 i$ are given by:

$\paren {-27 i}^{1/6} = \set {\sqrt 3 \, \map \cis {45 + 60 k} \degrees}$

for $k = 0, 1, 2, 3, 4, 5$.


That is:

\(\ds k = 0: \ \ \) \(\ds z = z_1\) \(=\) \(\ds \sqrt 3 \cis 45 \degrees\)
\(\ds k = 1: \ \ \) \(\ds z = z_2\) \(=\) \(\ds \sqrt 3 \cis 105 \degrees\)
\(\ds k = 2: \ \ \) \(\ds z = z_3\) \(=\) \(\ds \sqrt 3 \cis 165 \degrees\)
\(\ds k = 3: \ \ \) \(\ds z = z_4\) \(=\) \(\ds \sqrt 3 \cis 225 \degrees\)
\(\ds k = 4: \ \ \) \(\ds z = z_5\) \(=\) \(\ds \sqrt 3 \cis 285 \degrees\)
\(\ds k = 5: \ \ \) \(\ds z = z_6\) \(=\) \(\ds \sqrt 3 \cis 345 \degrees\)


Proof

Complex 6th Roots of -27 i.png


Let $z^6 = -27 i$.

We have that:

$z^6 = 27 \, \map \cis {\dfrac {3 \pi} 2 + 2 k \pi}$


Let $z = r \cis \theta$.

Then:

\(\ds z^6\) \(=\) \(\ds r^6 \cis 6 \theta\) De Moivre's Theorem
\(\ds \) \(=\) \(\ds 27 \, \map \cis {\dfrac {3 \pi} 2 + 2 k \pi}\)
\(\ds \leadsto \ \ \) \(\ds r^6\) \(=\) \(\ds 27\)
\(\ds 6 \theta\) \(=\) \(\ds \dfrac {3 \pi} 2 + 2 k \pi\)
\(\ds \leadsto \ \ \) \(\ds r\) \(=\) \(\ds 27^{1/6}\)
\(\ds \) \(=\) \(\ds \sqrt 3\)
\(\ds \theta\) \(=\) \(\ds \dfrac \pi 4 + \dfrac {k \pi} 3\) for $k = 0, 1, 2, 3, 4, 5$
\(\ds \theta\) \(=\) \(\ds 45 \degrees + k \times 60 \degrees\) for $k = 0, 1, 2, 3, 4, 5$

$\blacksquare$


Sources

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