The energy of E of a system is function of time t and is given by $E\left(t\right)=\alpha t-\beta t^3,$ where $\alpha$ and $\beta$ are constants. The dimensions of $\alpha$ and $\beta$ are

Energy of the system as a function of time  $E\left(t\right)=\alpha t-\beta t^3$. Where,  $\alpha and\beta$ are constant
According to principle of homogeneity,
Dimension of  $\alpha t$= dimension of energy
i.e
 $\Rightarrow \left[\alpha \right]\left[t\right]=\left[E\right] \Rightarrow \left[\alpha \right]\left[T\right]=\left[M{L}^2{T}^{{ }_{-2}}\right]$
 $\Rightarrow \left[\alpha \right]=\left[M{L}^2{T}^{{ }_{-3}}\right]$
Again, dimension of $\beta t^3=$ dimension of energy
$\Rightarrow \left[\beta \right]\left[t^2\right]=\left[M{L}^2{T}^{{ }_{-2}}\right]$

$\Rightarrow \left[\beta \right]\left[T^3\right]=\left[M{L}^2{T}^{{ }_{-2}}\right]\Rightarrow \left[\beta \right]=\left[M{L}^2{T}^{-5}\right]$