# Thread: Roots of an Argand Diagram

1. ## Roots of an Argand Diagram

I'm having trouble with a complex number roots question.
It provides a diagram, of a circle radius 2 and a line at 60 degrees.
It states the equation z^4 - a = 0
a) Find all the other roots
b) Determine 'a'
c) Determine z^4 - a = 0 exactly

Cheers

2. ## Re: Roots of an Argand Diagram

There are a number of things you did not say that must be true in order to make any sense of this. I presume that the problem also tells you that point where line and circle intersect is one solution to the equation. Further, I assume that when you say a "line at 60 degrees" you mean at 60 degrees to the x-axis. The four roots of $\displaystyle z^4= a$ lie on a circle of radius $\displaystyle |a|^{1/4}$ 90 degrees apart around the circle. You are told that one root lies on the line at 60 degrees (to the x-axis presumably) so the others lie on that same circle with angle at 60+90= 150 degrees, 60+180= 240 degrees, and 60+ 270= 330 degrees. Since the circle has radius 2, $\displaystyle |a|^{1/4}= 2$. You should be able to use simple trigonometry to find each of the four roots in "a+ bi" form.

3. ## Re: Roots of an Argand Diagram

Yes, your guesses are correct. So through that reasoning, the answer to part b) of the equation would be a = 16? Hence, making part c) z^4 - 16 = 0 ?