Question

find all the eighth root of (19+7i)

Answer 1

solution for this problem

Answer 2

A solution I will not provide, but I will give you everything you need to find it yourself.

Given the equation \(z^n=c=re^{it}\), you would use DeMoivre's theorem to take the \(n\)th root:
\[z=\left(re^{it}\right)^{1/n}=r^{1/n}e^{it/n}=r^{1/n}\left[\cos\left(t+\dfrac{2\pi k}{n}\right)+i\sin\left(t+\dfrac{2\pi k}{n}\right)\right]\]
where \(k=0,1,2,...,n-1\).

First thing to do is convert the given complex number into its trigonometric/polar form.
\[19+7i~~\implies~~\begin{cases}r=\sqrt{19^2+7^2}=\sqrt{410}\\\\
\theta=\tan^{-1}\dfrac{7}{19}\end{cases}\]
You're finding the eight roots, so \(n=8\).