A double star has two components of equal brightness, each with a magnitude of 8.34. if these stars are so close together that they appear to be one object, what is the apparent magnitude of the combined object

Assuming that the magnitude is additive, then you simply have to add the brightness four times. This is because each star has 2 magnitudes of 8.34. As a result, you have to add 4 magnitudes of 8.34. The solution is as follows:

Magnitude = 4 (8.34)

Magnitude = 33.36