${\mathrm{K}}_{\mathrm{p}}\text{ and }{\mathrm{K}}_{\mathrm{p}}^{\mathrm{\prime }}$ are the equilibrium constants of the two

reactions. given below  $\frac{1}{2}{\mathrm{N}}_{2\left(\mathrm{g}\right)}+\frac{3}{2}{\mathrm{H}}_{2\left(\mathrm{g}\right)}\rightleftarrows {\mathrm{NH}}_{3\left(\mathrm{s}\right)}$

${\mathrm{N}}_{2\left(g\right)}+3{\mathrm{H}}_{2\left(g\right)}\rightleftharpoons 2{\mathrm{NH}}_{3\left(\mathrm{g}\right)}$

Therefore. ${\mathrm{K}}_{\mathrm{p}}^{\mathrm{\prime }}\text{ and }{\mathrm{K}}_{\mathrm{p}}^{\mathrm{\prime }}$ are related by

When a balanced equation having equilibrium constant

${\mathrm{K}}_{\mathrm{P}}$ is multiplied by a certain value n, the equilibrium 

constant for the new equation will be equal to ${\left({\mathrm{K}}_{\mathrm{P}}\right)}^{\mathrm{n}}$.

In the given equation, $\mathrm{n}=2$

So, ${\mathrm{K}}_{\mathrm{P}}^{\mathrm{\prime }}={\left({\mathrm{K}}_{\mathrm{P}}\right)}^2\text{ or }{\mathrm{K}}_{\mathrm{P}}=\sqrt{{\mathrm{K}}_{\mathrm{P}}^{\mathrm{\prime }}}.$