## anonymous one year ago Find the fourth roots of the complex number z1= 1+ sqrt3 I Part 1:write z1 in polar form 2(cos60+isin60) Part 2: find the modulus of the roots of z1 I got 2 Part 3: find the four angles that define the fourth roots of the number z1 Part 4: what are the fourth roots of z1= sqrt3+1 i

• This Question is Open
1. anonymous

am I correct on part 1&2 and I don't know how to do 3&4

2. IrishBoy123

.

3. anonymous

@IrishBoy123 ??

4. anonymous

If $$z$$ has angle $$\theta$$, then the $$n$$th roots $$z^{1/n}$$ will follow a pattern of $$\dfrac{\theta+2k\pi}{n}$$, where $$k=0,1,\ldots,n-1$$.

5. anonymous

@SithsAndGiggles so if the angles i find are 15 and 60 the 4th roots would be $60+2k \div4$?

6. anonymous

Idk i just have no idea how to do this

7. anonymous

You found that $$\theta=60^\circ$$, right? In radians, that's $$\dfrac{\pi}{3}$$. Take $$k=0$$. Then the angle of the first (principal) fourth root is $$\dfrac{\dfrac{\pi}{3}+2\pi\times0}{4}=\dfrac{\pi}{12}$$, which in degrees is $$15^\circ$$. Now take $$k=1$$. This gives you an angle of $$\dfrac{\dfrac{\pi}{3}+2\pi\times1}{4}=\dfrac{7\pi}{12}$$, or $$105^\circ$$. Continue the pattern for $$k=2$$ and $$k=3$$.