Complexity!

$z={ i }^{ 1/2 }={e}^{i*\pi/4}=cos(\pi/4)+i*sin(\pi/4)$ .............[This is done by taking k=0 in the formula of complex numbers raised to the fractional powers]

Therefore, $z=1/\sqrt { 2 } +i*1/\sqrt { 2 }$

Therefore by the given question, $x*y=1/2$

$i = {(x+iy)}^2$
$i = ({x}^2) - ({y}^2) + 2ixy$

Comparing coefficients:

$2xy = 1 .......... 1$

$x^2 - y^2 = 0 ............ 2$

From equation 1 we can see that : $xy = 1/2$