# Find all the 8^(th) roots of 3i-3?

## Find all the ${8}^{t h}$ roots of $3 i - 3$ Also show any one of them in an Argand diagram.

Aug 23, 2017

#### Explanation:

Let $\omega = - 3 + 3 i$, and let ${z}^{8} = \omega$

First, we will put the complex number into polar form:

$| \omega | = \sqrt{{\left(- 3\right)}^{2} + {3}^{2}} = 3 \sqrt{2}$
$\theta = \arctan \left(\frac{3}{- 3}\right) = \arctan \left(- 1\right) = - \frac{\pi}{4}$
$\implies a r g \setminus \omega = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3 \pi}{4}$

So then in polar form we have:

$\omega = 3 \sqrt{2} \left(\cos \left(\frac{3 \pi}{4}\right) + i \sin \left(\frac{3 \pi}{4}\right)\right)$

We now want to solve the equation ${z}^{8} = \omega$ for $z$ (to gain $8$ solutions):

${z}^{8} = 3 \sqrt{2} \left(\cos \left(\frac{3 \pi}{4}\right) + i \sin \left(\frac{3 \pi}{4}\right)\right)$

Whenever dealing with complex variable equation such as this it is essential to remember that the complex exponential (and therefore the polar representation) has a period of $2 \pi$, so we can equivalently write (incorporating the periodicity):

${z}^{8} = 3 \sqrt{2} \left(\cos \left(\frac{3 \pi}{4} + 2 n \pi\right) + i \sin \left(\frac{3 \pi}{4} + 2 n \pi\right)\right) \setminus \setminus \setminus n \in \mathbb{Z}$

By De Moivre's Theorem we can write this as:

$z = 3 \sqrt{2} {\left(\cos \left(\frac{3 \pi}{4} + 2 n \pi\right) + i \sin \left(\frac{3 \pi}{4} + 2 n \pi\right)\right)}^{\frac{1}{8}}$
$\setminus \setminus = {\left(3 \sqrt{2}\right)}^{\frac{1}{8}} {\left(\cos \left(\frac{3 \pi}{4} + 2 n \pi\right) + i \sin \left(\frac{3 \pi}{4} + 2 n \pi\right)\right)}^{\frac{1}{8}}$
$\setminus \setminus = {3}^{\frac{1}{8}} {2}^{\frac{1}{16}} \left(\cos \left(\frac{\frac{3 \pi}{4} + 2 n \pi}{8}\right) + i \sin \left(\frac{\frac{3 \pi}{4} + 2 n \pi}{8}\right)\right)$
$\setminus \setminus = {3}^{\frac{1}{8}} {2}^{\frac{1}{16}} \left(\cos \theta + i \sin \theta\right)$

Where:

$\theta = \frac{\frac{3 \pi}{4} + 2 n \pi}{8} = \frac{\left(3 + 8 n\right) \pi}{32}$

And we will get $8$ unique solutions by choosing appropriate values of $n$. Working to 3dp, and using excel to assist, we get:

After which the pattern continues (due the above mentioned periodicity).

We can plot these solutions on the Argand Diagram: