\(\left[ {{M^1}{L^1}{T^{ - 2}}} \right]\,\, = \,\,{\left[ {{M^1}{L^{ - 1}}{T^2}} \right]^\alpha }\,{\left[ {L{T^{ - 1}}} \right]^\beta }{\left[ T \right]^\gamma }\)

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

\(\alpha \,\, = \,\,1\,;\,\, - \alpha \,\, + \;\,\beta \,\, = \,\,1\,\)

\(\beta \,\, = \,\,\,2,\,\,2 - 2\alpha \,\, - \,\,\beta \,\, + \;\,\gamma \,\, =  - 2\)

\(\therefore \,\, - 2\,\, - \,\,2\,\, + \;\,\gamma \,\, =  - 2\,\,\)

\(\,\gamma \,\, = \,\,2\)

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