# Find dimensional formula of $\alpha$ [duplicate]

The power delivered by a force is given by the relation $$P=\frac{\alpha}{\beta}e^{-\beta t},$$ where $t$ is time. Find the dimensional formula for $\alpha$.

So, $-\beta t$ doesn't make sense for the dimensional formula of $P$. But then we have two unknowns and one equation since we do not know if $\beta$'s dimensions. But then if I put $\beta t$ as dimensionless, it fits just right with the answer $([ML^2 T^{-4}])$ . But does this happen exactly. Why is $\beta t$ dimensionless, because we are just using it's numerical value. Please help. I don't understand it.

$\beta t$ must be dimensionless because it is the argument of a function (read this thread). Therefore
$$[\beta] = t^{-1}$$