Question

Find the fourth roots of the complex number " z1 = 1 + √3 * i

Answer 1

Part I: Write z1 in polar form.
Part II: Find the modulus of the roots of z1.
Part III: Find the four angles that define the fourth roots of the number z1.
Part IV: What are the fourth roots of the equation " z1 = 1 + √3 * i ".

Answer 2

Is my answer correct?

a = 1
b = sqrt(3)

sqrt(a^2 + b^2) = sqrt(1 + 3) = sqrt(4) = 2

z = 2 * (1/2 + i * sqrt(3)/2)
z = 2 * (cos(pi/3 + 2pi * k) + i * sin(pi/3 + 2pi * k))
z = 2 * (cos((pi/3) * (1 + 6k)) + i * sin((pi/3) * (1 + 6k)))
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6k)) + i * sin((pi/12) * (1 + 6k)))

2^(1/4) * (cos(pi/12) + i * sin(pi/12))
2^(1/4) * (cos(7pi/12) + i * sin(7pi/12))
2^(1/4) * (cos(13pi/12) + i * sin(13pi/12))
2^(1/4) * (cos(19pi/12) + i * sin(19pi/12))

Answer 3

looks good to me

Answer 4

modulus is 2

Answer 5

angle is \(\frac{\pi}{3}\)

Answer 6

you may verify the answers by rising the roots to 4th power
you should get back the z1

Answer 7

and \[\frac{\pi}{3}\times \frac{1}{4}=\frac{\pi}{12}\]

Answer 8

all looks swell

Answer 9

Should I simplify the roots?

Answer 10

@satellite73

Answer 11

i don't think you can simplify them further

Answer 12

I meant the angles.

Answer 13

they look good the way they are now

Answer 14

pi/12

what can you simplify here ?

Answer 15

@satellite73 said it simplifies to pi/3?

Answer 16

Nope, satellite was referring to something else

Answer 17

ok