According to the properties of the equilibrium constant, if a balanced chemical equation is multiplied by a factor '$n$', the new equilibrium constant becomes the original equilibrium constant raised to the power of '$n$' ($K' = K^n$). For the given reaction, the original equation $N_2(g) + O_2(g) \rightleftharpoons 2NO(g)$ is multiplied by $n = \frac{1}{2}$ to obtain $\frac{1}{2}N_2(g) + \frac{1}{2}O_2(g) \rightleftharpoons NO(g)$. Therefore, the new equilibrium constant will be $K^{1/2}$.