# Thread: Finding Solutions using Complex Numbers

1. ## Finding Solutions using Complex Numbers

I have been having trouble doing practice quesions on finding the solutions using complex numbers. I feel like I understand it on a very superficial level as I can solve simple ones, but I have trouble when I need to solve questions like z^5 = 12exp(i*pi/2) which shouldn't be too hard since it is already in polar form.
It would be greatly appreciated someone could give me a quick rundown on how to properly solve these types of questions or point me a good resource on them.

Edit: An example of what I am having trouble with:

z^2 = -7*sqrt(2)/2 + 7*sqrt(2)*i/2

Using the form z = r*exp(a*i), I find r to be 7 and a to be pi/4

So z = sqrt(7)exp(pi*n*i/8)

This is where I get stuck. The examples we did in class only involved then plugging 0, 1, 2, 3, etc until the results looped. However, the awnsers provided for this question are sqrt(7)exp(3*pi*i/8) and sqrt(7)exp(11*pi*i/8). What should I be doing instead?

2. ## Re: Finding Solutions using Complex Numbers

Originally Posted by Vanilla
I have been having trouble doing practice quesions on finding the solutions using complex numbers. I feel like I understand it on a very superficial level as I can solve simple ones, but I have trouble when I need to solve questions like z^5 = 12exp(i*pi/2) which shouldn't be too hard since it is already in polar form.
It would be greatly appreciated someone could give me a quick rundown on how to properly solve these types of questions or point me a good resource on them.

Edit: An example of what I am having trouble with:

z^2 = -7*sqrt(2)/2 + 7*sqrt(2)*i/2

Using the form z = r*exp(a*i), I find r to be 7 and a to be pi/4

So z = sqrt(7)exp(pi*n*i/8)

This is where I get stuck. The examples we did in class only involved then plugging 0, 1, 2, 3, etc until the results looped. However, the awnsers provided for this question are sqrt(7)exp(3*pi*i/8) and sqrt(7)exp(11*pi*i/8). What should I be doing instead?
There are exactly 2 distinct square roots, so plugging in 0 and 1 will produce both and beyond that they will just cycle through the two you already have.

Also:

$z^2=7e^{i(3\pi/4+2n\pi)}$

(that is $\arg(z)=3\pi/4$ )so:

$z=\sqrt{7}e^{i(3\pi/4+2n\pi)/2}=\sqrt{7}e^{i(\pi/8)(3+8n)}$

Now try plugging in values of $n$

CB