Explanation

Given that fundamental quantities are momentum (p), area (A) and time (T). Let us consider the dimensional formula for $E \propto [p^{a}A^{b}T^{c}]$ $E = [kp^{a}A^{b}Tc]$ where k is dimensionless constant of proportionality. Dimensions of energy $[E] = [ML^{2}T^{-2}]$ and Dimension of momentum $p = mv = [MLT^{-1}]$. Dimension of Area $[A] = [L^{2}]$. Dimension of Time $[T] = [T]$ $ML^{2}T^{-2} = [MLT^{-1}]^{a}[L^{2}]^{b}[T^{c}] = M^{a}L^{2b+a}T^{-a+c}$ By principle of homogeneity of dimensions, $a = 1$, $2b + a = 2 \Rightarrow b = 1/2$, $-a + c = -2 \Rightarrow c = -1$ So, Dimensional formula (of energy) $E = [pA^{1/2}T^{-1}]$