$F\,\, = \,\,{P^\alpha }{v^\beta }{T^\gamma }$

$\left[ {{M^1}{L^1}{T^{ – 2}}} \right]\,\, = \,\,{\left[ {{M^1}{L^{ – 1}}{T^2}} \right]^\alpha }\,{\left[ {L{T^{ – 1}}} \right]^\beta }{\left[ T \right]^\gamma }$

$\,\,\left[ {{M^1}{L^1}{R^{ – 2}}} \right]\,\, = \,\,\left[ {{M^\alpha }{L^{ – \alpha \, + \;\beta }}{T^{ – 2\alpha  – \beta  + \gamma }}} \right]$

$\alpha \,\, = \,\,1\,;\,\, – \alpha \,\, + \;\,\beta \,\, = \,\,1\,$

$\beta \,\, = \,\,\,2,\,\,2 – 2\alpha \,\, – \,\,\beta \,\, + \;\,\gamma \,\, =  – 2$

$\therefore \,\, – 2\,\, – \,\,2\,\, + \;\,\gamma \,\, =  – 2\,\,$

$\,\gamma \,\, = \,\,2$

$ \Rightarrow \,\,F\,\, = \,\,P{v^2}{T^2}$