Final answer:
The ratio of brightness between two stars with apparent magnitudes of 0.06 and 1.06 is 2.5, meaning the brighter star (magnitude 0.06) is 2.5 times brighter than the other star (magnitude 1.06).
Step-by-step explanation:
The question at hand is to calculate the ratio of brightness between two stars with apparent magnitudes of 0.06 and 1.06. To find this ratio, we use the formula that a difference of 5 magnitudes corresponds to a brightness ratio of 100:1. Since a single point of magnitude is equivalent to the 5th root of 100, or approximately 2.5, the magnitude difference of 1 (1.06 - 0.06) means we need to calculate the brightness difference for a single magnitude.
Using the rule that each step of 1 magnitude corresponds to a factor of 2.5 in brightness, the brightness ratio of two stars with a one magnitude difference is thus 2.5. Therefore, the star with an apparent magnitude of 0.06 is 2.5 times brighter than the star with an apparent magnitude of 1.06.