Need help in solving for the roots of this biquadratic equation

Question

I was trying to make random polynomials and solve for their roots when I came across this. $$x^4 - \sqrt{2}$$
I tried solving for the roots by equating the above expression to zero and I got this: $$x = 2^{1/8}$$ How do I find the other roots for this equation? Are all roots equal in this case?

Please explain as elaborately as possible.

I am confused, please help me out. Thanks in advance.

Answer 1

The roots are just $\sqrt[8]{2}\omega$, where $\omega$ is an fourth root of unity, because $(\sqrt[8]{2}\omega)^4=\sqrt{2}\omega^4=\sqrt{2}\cdot 1=\sqrt{2}$. Since there are four different fourth roots of unity: $\pm 1, \pm i$ we just got four solutions, only two in real numbers.