If force $\left[F\right]$, acceleration $\left[A\right]$ and time $\left[T\right]$ are chosen as the fundamental physical quantities. Find the dimensions of energy.

Let consider,

$\left[\mathrm{Energy}\right]=\left[\mathrm{F}{]}^{\mathrm{a}}\right[\mathrm{A}{]}^{\mathrm{b}}[\mathrm{T}{]}^{\mathrm{c}}$

Now put the dimensional formula of the given quantities,

$\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]={\left[{\mathrm{MLT}}^{-2}\right]}^{\mathrm{a}}{\left[{\mathrm{LT}}^{-2}\right]}^{\mathrm{b}}[\mathrm{T}{]}^{\mathrm{c}}$

$\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]=\left[{\mathrm{M}}^{\mathrm{a}}{\mathrm{L}}^{\mathrm{a}+\mathrm{b}}{\mathrm{T}}^{-2\mathrm{a}-2\mathrm{b}+\mathrm{c}}\right]$

On comparing powers of $\left[M\right]$

$a=1$

On comparing powers of $\left[L\right]$

$a+b=2$

$\Rightarrow 1+b=2$

$\Rightarrow b=1$

On comparing powers of $\left[T\right]$

$-2 a-2 b+c=-2$

$\Rightarrow -2-2+c=-2$

$\Rightarrow c=2$

$[ Energy ]=\left[F\right]\left[A\right]\left[T^2\right]$