by **Baltuilhe** » Tue Sep 21, 2021 10:00 pm

Sample:

[tex]z=a+bi[/tex]

or

[tex]z=\rho\cdot\left(\cos\theta+i\sin\theta\right)[/tex]

So:

[tex]a=\rho\cos\theta\\

b=\rho\sin\theta[/tex]

and

[tex]\tan\theta=\frac{b}{a}\\\\

\rho=\sqrt{a^2+b^2}[/tex]

De Moivre's Root Theorem:

[tex]z^{1/n}=\rho^{1/n}\cdot\left[\cos\left(\frac{\theta+2k\pi}{n}\right)+i\sin\left(\frac{\theta+2k\pi}{n}\right)\right][/tex]

k=0,1,2,...,n-1

Numeric:

[tex]z=16\cdot\left[\cos\left(\dfrac{\pi}{6}\right)+i\sin\left(\dfrac{\pi}{6}\right)\right][/tex]

Fourth roots:

[tex]z^{1/4}=16^{1/4}\cdot\left[\cos\left(\dfrac{\dfrac{\pi}{6}+2k\pi}{4}\right)+i\sin\left(\dfrac{\dfrac{\pi}{6}+2k\pi}{4}\right)\right][/tex]

[tex]z^{1/4}=(2^4)^{1/4}\cdot\left[\cos\left(\dfrac{\pi+12k\pi}{24}\right)+i\sin\left(\dfrac{\pi+12k\pi}{24}\right)\right][/tex]

k=0

[tex]z^{1/4}=2\cdot\left[\cos\left(\dfrac{\pi}{24}\right)+i\sin\left(\dfrac{\pi}{24}\right)\right][/tex]

k=1

[tex]z^{1/4}=2\cdot\left[\cos\left(\dfrac{13\pi}{24}\right)+i\sin\left(\dfrac{13\pi}{24}\right)\right][/tex]

k=2

[tex]z^{1/4}=2\cdot\left[\cos\left(\dfrac{25\pi}{24}\right)+i\sin\left(\dfrac{25\pi}{24}\right)\right][/tex]

k=3

[tex]z^{1/4}=2\cdot\left[\cos\left(\dfrac{37\pi}{24}\right)+i\sin\left(\dfrac{37\pi}{24}\right)\right][/tex]