find all the seventh roots of (3+4i)
First, convert the complex number to polar form
\( 3 + 4 i=5 \cos{\left({atan}{\left(\frac{4}{3} \right)} \right)} + 5 i \sin{\left({atan}{\left(\frac{4}{3} \right)} \right)} \)
According to the De Moivre’s Formula, all n-th roots of a complex number
\( r\left(\cos\left(\theta\right)+i\sin\left(\theta\right)\right) \)
are given by \( \sqrt[n]{r}\left(\cos\left(\frac{\theta+2\pi k}{n}\right)+i\sin\left(\frac{\theta+2\pi k}{n}\right)\right) \)
We have that \( r = 5, \theta= {atan}{\left(\frac{4}{3} \right)}, n=7 \)
Sub in k = 0, 1, 2, 3, 4, 5, 6 to get all of the roots in polar form.
\( \sqrt[7]{3 + 4 i}=\sqrt[7]{5} \cos{\left(\frac{{atan}{\left(\frac{4}{3} \right)}}{7} \right)} + \sqrt[7]{5} i \sin{\left(\frac{{atan}{\left(\frac{4}{3} \right)}}{7} \right)}\approx 1.24747270589553 + 0.166227124177353 i \)