What does a supernova look like at its peak luminosity? I know that in some types of supernovae, the cause of the increased luminosity is the radioactive decay of certain elements ejected during the explosion, so a question came to my mind.
If the ejected material carrying the isotopes that decay to give the electromagnetic radiation is expelled at velocity of say 5% the speed of light, and given the fact that some supernovae stay extremely luminous for more than 4 weeks, then by that time the radioactive isotopes will have traveled more than 30 billion kilometers from the exploded star.
So, does that mean a supernova at 4 weeks can be expected to look like a star with a radius of 30 billion kilometers and luminosity of $10^8$-$10^9$ times the solar luminosity ?
Or I am getting the idea of the radioactive decay as the source of the supernova luminosity wrong ?
 A: Your math does check out:
\begin{align}
r&=vt \\
 &=0.05\cdot2.9979\times10^{10}\frac{cm}s\cdot4\cdot604800\,s\\
 &=3.63\times10^{15}\,cm\\
&=36.3\times10^9\,km\\
&=0.012\,pc
\end{align}
When a supernova explodes, it enters the free expansion phase, it's position is linear in time ($r=vt$, as used above). It stays in this phase for a few hundred years (depends heavily on the ambient density); assuming 200 years, then
$$
r_{fe}=9.45\times10^{18}\,cm=3\,pc
$$
After this point, the Supernova Remnant (technically speaking, SNe is the explosion while SNR is the result of the material after said explosion) continues expanding, though at a reduced rate (because it has swept-up ambient material this entire time, building up a thick shell of thickness $w\sim0.1\,pc$) for many thousands of years.
SNe theory says a normal Type Ia produces about $0.5\,M_\odot$ of nickel-56 which then decays to an excited state of cobalt-56, which then emits an X-ray photon:
\begin{align}
\,^{56}{\rm Ni}+e^-&\to\,^{56}{\rm Co}^*+\nu_e \\
\,^{56}{\rm Co}^*&\to\,^{56}{\rm Co}+\gamma
\end{align}
The cobalt-56 (lifetime around 100 days) then decays to iron-56 which also decays with some X-ray photons. Until SN 2014J, we had only observed the iron-56 decay line due to the fact that the lifetime of the above reaction is about 9 days and the ejecta are opaque to these lines due to Compton scattering in this same time-frame. SN 2014J provided $\gamma$-ray and X-ray emissions due to the cobalt-56, proving the theory correct.
Note that the shell remains very thick during this whole time. Wikipedia provides an image of SN 1006 (exploded in the year 1006, so it's now 1008 years old) that shows the expansion of the shell:

This shell is measured to be between 0.04 and 0.2 pc, which are roughly $1.2\cdot10^{12}$ km and $6.2\cdot10^{12}$ km thick, which is just shy of 1 lightyear. And after all this time, it is strong in radio, X-ray & $\gamma$-ray emissions (from this site):

