Is there a fast and easy way to solve $x^7-10=0$?

Question

I came across the polynomial equation $$x^7-10=0$$ and by using a calculator, i found its solutions to be very interesting, in fact all of them are of the form $$x=\sqrt[7]{10}e^{\pm \frac{2k i \pi}7}$$ for $k=0,1,2,3$$. So, first, i would like to know if there is anything special about this polynomial? And second, can we actually solve it with using a faster method than polynomial division?

Answer 1

This is due to $\exp(2n \pi i)=\cos(2n \pi ) + i \sin (2n \pi)=1$, $$x^7 = 10=10\exp(2n\pi i)$$

Hence

$$x=10^{\frac17}\exp\left(\frac{2n \pi i}7 \right)$$

If we draw the argand diagram, the solutions fall on the circle centered at origin with radius $10^\frac{1}{7}$ and evenly spread out and one of the point is positive.