Finding Fourth Roots of -2√3 + i2

[kring_c14]

Homework Statement

find the four fourth roots of -2$$\sqrt{3}$$+i2


i don't have any attempt for a solution because i don't know what to do..
im really lost.. i regret sleeping in class

 

[HallsofIvy]

I imagine you were intended to use DeMoivres' theorem:
If a complex number can be written in polar form $z= r(cos(\theta)+ i sin(\theta))$ then its nth power, zⁿ, can be written $z^n= r^n(cos(n\theta)+ i sin(n\theta)$
In your case, n is the fraction 1/4. Convert $-2\sqrt{3}+ 2i$ to polar form (which happens to be pretty simple). Take the real fourth root of r. Remember that you can add any multiple of $2\pi$ to $\theta$. Dividing by 1/4 will give you different results for different multiples of $2\pi$.

 

[kring_c14]

im having some problem in the angle..
what i dis is this

z=r cis (theta)
x=-2(sqrt3)
y=2
r=4
so
theta=-60

then, will i just substitute the numbers to the equation?

 

[HallsofIvy]

Yes, of course. r= 4 and theta= - 60 degrees (although I would prefer theta= -$\pi/3$).

 

[kring_c14]

i think the angle is is -30...

soln:

x=-2(sqrt3)
y=2
r=4
tan (theta)= 2/[-2(sqrt3)]
=-1/sqrt3=-30degrees=-pi/6

shouldn't I make the angle positive?

if yes
should i subtract 30 from 180
or subtract 30 from 360?

i'm totally clueless...
desperately needing some help

 

[kring_c14]

pls pls pls...help me with this one..can anyone give me a complete solution for this?thanks

 

[Dick]

It looks to me like your angle is more like 150 degrees. You are in the second quadrant. So yes, subtract it from 180.