Final answer:
Star A with an apparent magnitude of 1 is about 22,627 times brighter in apparent brightness than star B with an apparent magnitude of 14, calculated by raising 2.5 to the power of the magnitude difference (2.5^13).
Step-by-step explanation:
According to the magnitude system established in the nineteenth century, a difference of 5 magnitudes in apparent brightness corresponds to a brightness ratio of 100:1. Having a difference of 13 magnitudes between star A with an apparent magnitude of 1 and star B with an apparent magnitude of 14, we can calculate the brightness difference. We use the fact that each magnitude difference of 1 corresponds to a brightness ratio of approximately 2.5 times. Therefore, to find how many times brighter star A is compared to star B, we raise 2.5 to the thirteenth power.
The calculation would be 2.5^13 which equals approximately 22,627. Therefore, star A is about 22,627 times brighter in apparent brightness than star B.