If equilibrium constants of reaction, ${\mathrm{N}}_2+{\mathrm{O}}_2\rightleftharpoons 2\mathrm{NO}$ is  ${\mathrm{K}}_1$ 

and $\frac{1}{2}{\mathrm{N}}_2+\frac{1}{2}{\mathrm{O}}_2\rightleftharpoons \mathrm{NO}\text{ is }{\mathrm{K}}_2$, then

${\mathrm{N}}_2+{\mathrm{O}}_2\rightleftharpoons 2\mathrm{NO}$           ..(i)

$\frac{1}{2}{\mathrm{N}}_2+\frac{1}{2}{\mathrm{O}}_2\rightleftharpoons \mathrm{NO}$           …(ii)

For equation number (i) 

${\text{K}}_1=\frac{[\text{NO}{]}^2}{\left[{\mathrm{N}}_2\right]\left[{\mathrm{O}}_2\right]}$                   …(iii)

For equation number (ii) 

${\text{K}}_2=\frac{\left[\mathrm{NO}\right]}{{\left[{\text{N}}_2\right]}^{1/2}{\left[{\mathrm{O}}_2\right]}^{1/2}}$         …(iv)

From equation (iii) and (iv) it is clear that

${\text{K}}_2={\left({\mathrm{K}}_1\right)}^{1/2}=\sqrt{{\mathrm{K}}_1};{\text{ Hence, K}}_2=\sqrt{{\mathrm{K}}_1}$