= Solution
The dimensions of <mass density>, <time> and <energy> are $\mathsf M\mathsf L^{-3}$, $\mathsf T$ and $\mathsf M\mathsf L^2\mathsf T^{-2}$. Matching powers in the proposed <dimensional analysis> gives
$$
\mathsf L=\mathsf M^{\alpha+\gamma}\mathsf L^{-3\alpha+2\gamma}\mathsf T^{\beta-2\gamma},\qquad
\alpha+\gamma=0,\quad -3\alpha+2\gamma=1,\quad \beta-2\gamma=0.
$$
Thus
$$
\boxed{\alpha=-\frac15,\qquad\beta=\frac25,\qquad\gamma=\frac15,\qquad R=C\left(\frac{Et^2}{\rho}\right)^{1/5}.}
$$
This is the <Sedov-Taylor blast wave> scaling for a fixed released <energy>. <Dimensional analysis> cannot determine the dimensionless $C$; that requires the fluid equations and thermodynamic assumptions. The assumed list of controlling parameters excludes, for example, an independent ambient sound speed or a continuing energy supply.
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