The star Procyon has a parallax of .286" and an apparent magnitude of .5. What is the number of times it is brighter/dimmer than it would be at 10...

Hello!

By the definition, 1 parsec is the distance from which 1 astronomical unit has angular size of 1 arcsecond (denoted 1"). This is the same as to have parallax of 1". And the more the distance, the less (proportionally) its corresponding parallax:

(parallax in angle seconds) * (distance in parsecs) = 1.

For Procyon, the parallax of 0.286" means the distance 1/0.286 = 10/2.86 parsecs, or approximately 3.5 parsecs.

If we imagine that Procyon becomes 10 parsecs from Earth, it would be

`(10)/(10/2.86) = 2.86`  

times farther than now. Its apparent (visible) brightness would be lower than now with this coefficient squared:

`(2.86)^2 approx 8.18.`

This is the answer: now Procyon is approximately 8.18 times brighter than if it would be at 10 parsecs from Earth.

That said, 10 parsecs is the conventional distance to measure the absolute magnitude.