$\displaystyle i = cos(\frac{\pi}{2} ) + i sin(\frac{\pi}{2})$,

$\displaystyle i^{1/2} = {(cos(\frac{\pi}{2} ) + i sin(\frac{\pi}{2})) }^{1/2}$,

$\displaystyle i^{1/2} = cos(\frac{\pi}{4} ) + i sin(\frac{\pi}{4})$,

$\displaystyle i^{1/2} = \frac{1}{\sqrt{2}}(1+i)$.

By the way, have you learned the

cyclotomic polynomial?

If so, then you'll see that $\displaystyle \Phi_8(x)=x^4+1$ and $\displaystyle \Phi_8(x)= (x -\zeta )(x - \zeta^3)(x-\zeta^5)(x-\zeta^7)$, where $\displaystyle \zeta = \frac{1}{\sqrt{2}}(1+i)$.