# Thread: one fourth root of w is 2+3j what are the rest

1. ## one fourth root of w is 2+3j what are the rest

hello again,

and thanks in advance for any hint/help

the moduli of the root i am given is 13^1/2 so the moduli of w is (13^1/2 )^4

what I am having difficulty with is finding the argument of w. the argument of the root i am given is arctan(3/2) = 56,309

and the forumla linking this with the argument of w is 56,309 = (2kpi + arg(w))/4, which is true for a specific value of k out of the possible values of k, i.e. 0,1,2,3.

x

2. If x^4=y what does (x*i)^4 equal?
That should give you a big clue.
A similar trick works for any other kind of root, though it is only quite this easy for fourth roots. Once you've understood this one try it for cube roots noting that the cube roots of 1 are 1,w,w^2 where w=-0.5 + sqrt(-0.75) (or its square).

4. Originally Posted by pb6883
(2kpi + arg(w))/4, which is true for a specific value of k out of the possible values of k, i.e. 0,1,2,3.
Suppose that we let $\displaystyle \xi = \arg (w) = \arctan \left( {\frac{3}{2}} \right)$ then $\displaystyle w = \sqrt {13} e^{i\xi }$.
That is the one given root. The other three are: $\displaystyle \sqrt {13} e^{i\left( {\xi + \frac{{k\pi }}{2}} \right)} \;;\,k = 1,2,3$.

5. is it arbitrary to take the given root as that for which k is zero??

6. Originally Posted by pb6883
is it arbitrary to take the given root as that for which k is zero??
That is the sort of question we all hate to get in class.
The answer can do more harm than good.
So let me explain what is really going with roots.
The four fourth roots are equally spaced on a circle at the origin with radius equal to the forth root of the absolute value.
So the angle ‘between’ any to roots is $\displaystyle \frac{\pi}{2}$.
Therefore adding $\displaystyle \frac{k\pi}{2}~,~k=1,2,3$ to the argument of any of the roots will always give the same set as we 'move' about the circle.
Do you see that?

7. yes i do, that really makes much sense... i suppose it will be good to practice the questions on diagrams as well as algebraicly to get used to exactly that sort of thing