Adjective or proper noun making nouns unique adn specific

Hello,

I was studying and doing an exercise on articles in the online grammar site of The Writing Center at Rensselaer Polytechnic Institute and came upon this explanation. I have omitted some parts of the explanations for brevity. Can you explain more about this? Sorry for not writing the whole excercise article out but please help me as best as you can.

Roman empire:

the -- singular; definite becasue of the preceding adjective: Roman. "Roman empire" is very specific (especially since "Roman" is derived from the proper noun, "Rome")

Roman Conquest??

the -- Conquest in this context is uncountable, meaning "the area or territory which was conquered." Because the preceding adjective, Roman, is derived from a proper name (Rome), it makes the following noun unique in this context

Also, further down in the exercise, the person wrote this about the word 'strength. Can you please tell me more about this explanation. -- how it could be like that?

Strength to overcome other problems:

the or 0 -- Strength can be either uncountable (the abstract quality or strength) or singular ( a particular instance of that quality). So it is either singular or defintie because of reason 4 (followinig modification: to overcome other problems), or uncountable and indefinite. Both would be equally acceptable, so it just depends on how thw writer is thinking.