Finding the roots of $x^5-2$. My question is about finding the $5^{th}$ roots of 2.While the associated question is about finding roots of unity

Question

I know the roots of polynomial $x^5-2$ are $\sqrt[5]{2},\zeta_{5}^{i}\sqrt[5]{2}$, where $0<i<5$, but I don’t know how we get these roots except $\sqrt[5]{2}$. Can someone guide me the method of finding these roots?

Answer 1

Well, you can think of $2$ as $2e^{2k\pi i}$, with $k\in \mathbb{Z}$. Then if $x^5-2=0$ we have $$ x^5=2 e^{2k\pi i} $$ and so $$ (x^5)^{1/5}=(2 e^{2k\pi i})^{1/5}. $$

This way $$ x=\sqrt[5]{2}e^{\frac{2k\pi i}{5}} $$

Finally, you have to think about how many different numbers you can get when you change $k\in \mathbb{Z}$. It turns out that the only different case are $k=0,1,2,3,4$.