# Thread: compelx roots

1. ## compelx roots

hello, need help on another problem. On my review sheet we got one problem that asks "Find all complex roots of the equation. Write the answer in algebraic form. X^4 = i" I remember how to start this but im not sure where to go from there... we learned this some time ago.

This is what I have:

X^4 = i
[r^4, 4(angle)] = i [this is putting it into polar form so we can find the answers in the complex plane]

from here i know that the answer is represented as a circle and that there are 4 solutions for x but not quite sure where to go from here.

Thanks for any help that can be given.

2. Hello
Originally Posted by rm1991
This is what I have:

X^4 = i
[r^4, 4(angle)] = i [this is putting it into polar form so we can find the answers in the complex plane]

from here i know that the answer is represented as a circle and that there are 4 solutions for x but not quite sure where to go from here.
Using the polar form, $X=r\exp(i\theta)$.

$r^4\exp(4i\theta)=i \Longleftrightarrow \left\{
\begin{array}{l}
|r^4\exp(4i\theta)|=|i|\\
\arg(r^4\exp(4i\theta))\equiv \arg(i)\, [2\pi]
\end{array}\right.
\Longleftrightarrow \left\{
\begin{array}{l}
r^4=1\\
4\theta \equiv \frac{\pi}{2} \,[2\pi]
\end{array}\right.$

Does it help ?

3. Im not familiar with this form (im in precalc)

rexp(i angle ) is just the polar form? or r^(power)cis (power * angle)?

also not familiar with arg( )...

4. Originally Posted by rm1991
Im not familiar with this form (im in precalc)

rexp(i angle ) is just the polar form?
Yes, I edited my post.

also not familiar with arg( )...
$\arg \left(r\exp(i \times \mathrm{angle})\right) = \mathrm{angle}$, that's all. What I wrote in the previous post is just that if $r_1\exp(i\theta_1)=r_2\exp(i\theta_2)$ then $r_1=r_2$ and $\theta_1\equiv \theta_2\,[2\pi]$.