Final answer:
The faintest stars visible to the nak'd eye have an apparent magnitude of about 6. The apparent brightness of stars is measured on a logarithmic scale, with each step in magnitude representing a change in brightness by a factor of approximately 2.5.
Step-by-step explanation:
The faintest stars visible to the nak'd eye in the night sky have an apparent magnitude of roughly 6. This value is based on a system where a first-magnitude star is one of the brightest stars visible and a sixth-magnitude star is one of the faintest stars that can just barely be seen without the aid of telescopes. A star with an apparent magnitude of 6 is right at the limit of human visual perception in a very dark and clear sky. As the magnitude number increases, the brightness decreases, meaning higher magnitude numbers correspond to fainter stars.
Regarding the apparent magnitudes and their differences in brightness, the magnitude system is logarithmic. A difference of 5 magnitudes equates to a brightness difference of a factor of 100. Therefore, each magnitude step represents approximately a difference of a factor of 2.5 in brightness. For example, a magnitude 16 star is much less bright than Antares, which has a magnitude of roughly 1. The exact calculation would involve finding the difference in magnitude (16 - 1 = 15), and then calculating how many times fainter it is based on the magnitude scale.