Roots of Complex Number/Examples/4th Roots of -16 i

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Example of Roots of Complex Number: Corollary

The complex $4$th roots of $-16 i$ are given by:

$\paren {-16 i}^{1/4} = \set {2 \, \map \cis {67.5 + 90 k} \degrees}$

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


That is:

\(\ds k = 0: \ \ \) \(\ds z = z_1\) \(=\) \(\ds 2 \cis 67.5 \degrees\)
\(\ds k = 1: \ \ \) \(\ds z = z_2\) \(=\) \(\ds 2 \cis 157.5 \degrees\)
\(\ds k = 2: \ \ \) \(\ds z = z_3\) \(=\) \(\ds 2 \cis 247.5 \degrees\)
\(\ds k = 3: \ \ \) \(\ds z = z_4\) \(=\) \(\ds 2 \cis 337.5 \degrees\)


Proof

Complex 4th Roots of -16 i.png


Let $z^4 = -16 i$.

We have that:

$z^4 = 16 \, \map \cis {\dfrac {3 \pi} 2 + 2 k \pi}$


Let $z = r \cis \theta$.

Then:

\(\ds z^4\) \(=\) \(\ds r^4 \cis 4 \theta\) De Moivre's Theorem
\(\ds \) \(=\) \(\ds 16 \, \map \cis {\dfrac {3 \pi} 2 + 2 k \pi}\)
\(\ds \leadsto \ \ \) \(\ds r^4\) \(=\) \(\ds 16\)
\(\ds 4 \theta\) \(=\) \(\ds \dfrac {3 \pi} 2 + 2 k \pi\)
\(\ds \leadsto \ \ \) \(\ds r\) \(=\) \(\ds 16^{1/4}\)
\(\ds \) \(=\) \(\ds 2\)
\(\ds \theta\) \(=\) \(\ds \dfrac {3 \pi} 8 + \dfrac {k \pi} 2\) for $k = 0, 1, 2, 3$
\(\ds \theta\) \(=\) \(\ds 67.5 \degrees + k \times 90 \degrees\) for $k = 0, 1, 2, 3$

$\blacksquare$


Sources

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