Question

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

Answer 1

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

Answer 2

.

Answer 3

@IrishBoy123 ??

Answer 4

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

Answer 5

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

Answer 6

Idk i just have no idea how to do this

Answer 7

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 \(k=3\).