$\left[ P \right]=\frac{N.m}{s}=\frac{kg.\frac{m}{{{s}^{2}}}.m}{s}=\frac{kg.{{m}^{2}}}{{{s}^{3}}}$
$kg{{(\frac{m}{s})}^{2}}\frac{1}{s}\xrightarrow{\left[ v \right]=\left[ \frac{1}{\sqrt{{{\varepsilon }_{{}^\circ }}{{\mu }_{{}^\circ }}}} \right]=\left[ \frac{m}{s} \right]}\left[ P \right]=\frac{kg}{m}\times \left[ \frac{1}{{{\varepsilon }_{{}^\circ }}{{\mu }_{{}^\circ }}} \right]\times \frac{m}{s}\Rightarrow \left[ P \right]=\left[ \mu \frac{1}{{{\varepsilon }_{{}^\circ }}{{\mu }_{{}^\circ }}}-\frac{1}{\sqrt{{{\varepsilon }_{{}^\circ }}{{\mu }_{{}^\circ }}}} \right]\equiv \left[ \mu {{\varepsilon }_{{}^\circ }}^{-\frac{3}{2}}{{\mu }_{{}^\circ }}^{-\frac{3}{2}} \right]$
$\alpha +\beta +\gamma =1-\frac{3}{2}-\frac{3}{2}=-2$