= Solution
The <Doppler factors> for the two discrete blobs are $\mathcal D_\pm=[\gamma(1\mp\beta\cos\theta)]^{-1}$. Their ratio is $(1+x)/(1-x)=77/33=7/3$. <Lorentz-invariant specific intensity> means $I_\nu/\nu^3$ is unchanged; with $\nu=\mathcal D\nu'$ and the same rest-frame <power-law spectrum>,
$$
I_\nu(\nu)=\mathcal D^3I'_{\nu'}(\nu/\mathcal D)
\propto\mathcal D^{3+\alpha}\nu^{-\alpha}.
$$
The brightness ratio at the common observed <frequency> therefore gives
$$
24=\left(\frac73\right)^{3+\alpha},\qquad
\boxed{\alpha=\frac{\ln24}{\ln(7/3)}-3=0.75081.}
$$
The stated identical intrinsic <specific intensities> must refer to comparable intrinsic states. If the blobs fade with their proper ages, equal observer times sample different ages and need that correction. A steady continuous <jet> has a different boosting exponent, $2+\alpha$; the discrete ejecta and the specified <specific intensity> transformation justify $3+\alpha$ here.
For <optically thin> <synchrotron radiation> from an isotropic <electron> distribution $dN/dE\propto E^{-p}$, the characteristic emitted <frequency> scales as $BE^2$ and the total single-particle <power> as $B^2E^2$. Thus $j_\nu\propto E^{-p}E^2/(d\nu/dE)\propto\nu^{-(p-1)/2}$; integration over the full single-particle spectrum preserves this exponent. Hence
$$
\boxed{p=2\alpha+1\simeq2.50.}
$$
The test-particle strong nonrelativistic <diffusive shock acceleration> prediction is $p=(r+2)/(r-1)=2$ for compression ratio $r=4$, giving $\alpha=0.5$. The inferred spectrum is steeper. If interpreted literally within that uncooled formula, $p=2.50$ would correspond to $r\simeq3$, but cooling, escape, and other acceleration conditions can also steepen it.
The observed Galactic <cosmic ray> differential <energy> spectrum below the knee is approximately $E^{-2.7}$, steeper still; its formal <synchrotron radiation> equivalent would be $\alpha\simeq0.85$. Thus $p\simeq2.50$ lies between the simplest source prediction and the propagated <cosmic ray> slope. The comparison is qualitative: most measured <cosmic rays> are nuclei, whereas the radio slope here probes <electrons>, and propagation and radiative losses alter the spectra differently.
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