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A typical quasar is believed to be approximately the size of the solar system, with a power output similar to that of a thousand galaxies.
Estimate, with reference to the inverse-square law, how much further the most distant visible quasar is likely to be compared to the most distant visible galaxy.
Quasars are very bright objects, that are some of the most distant astronomical bodies ever observed. They are a type of active galactic nuclei (AGN) which are very bright, and powerful galaxies, thought to be powered by matter falling into a supermassive black hole. The matter in the accretion disk is heated, through, friction to extremely high temperatures. So high, if fact, that a broad range of EM rays, up to x-rays can be emitted. This radiation is often collimated, or directed at right angles to the rotational axis of the black hole. If this beam is directed towards our own Earth it is observed as a quasar. As stated in the question, these objects can produce as much power as thousands of galaxies, maxing them the brightest objects in the universe. They are all typically a long way from our part of the universe, and are never observed in local galaxies, this implies that they were common in the younger universe, but very rare now.
In the question above we are asked to make an estimate on the distance to distant quasars, but are given very little information other than the quasar is $1000 \times$ brighter than a galaxy and that we should use the inverse square law to make our estimate.
The intensity ($I$) in this question is the intensity of the object from Earth, which is equivalent to the apparent magnitude. The luminosity ($L$) is the power output of the object. If the quasar is $\times 1000$ as luminous as a galaxy, but with the same apparent magnitude (which would be only just visible through whatever telescope was being used) or intensity we can apply the inverse square law as:
Intensity of galaxy=Intensity of quasar
\begin{align} \frac{1000\times L_{G}}{4π{d_{Q}}^{2}}&=\frac{L_{G}}{4π{d_{G}}^{2}}\\ \\ \frac{1000}{{d_{Q}}^{2}}&=\frac{1}{{d_{G}}^{2}} \end{align}So the distance to the most distant visible quasar would be:
So according to this the most distant quasar would be $\times 32$ further away than the furthest galaxy.
However in reality with the development of the space telescopes and large terrestrial optical telescopes, and visual processing the most distant galaxy ever observed is $\quantity{13.4\times 10^{9}}{ly}$ from Earth and the most distant quasar is $\quantity{13.1\times 10^{9}}{ly}$, however, these observed objects are right at the edge of the observable universe, and are also some of the oldest objects in the universe. More typically it is true that quasars are easier to see at great distances than galaxies.