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

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

Mathematics
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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 ".
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))
looks good to me

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Other answers:

modulus is 2
angle is \(\frac{\pi}{3}\)
you may verify the answers by rising the roots to 4th power you should get back the z1
and \[\frac{\pi}{3}\times \frac{1}{4}=\frac{\pi}{12}\]
all looks swell
Should I simplify the roots?
i don't think you can simplify them further
I meant the angles.
they look good the way they are now
pi/12 what can you simplify here ?
@satellite73 said it simplifies to pi/3?
Nope, satellite was referring to something else
ok

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