Before this is marked as a duplicate I just want to say that I've already read a similar thread, where the original poster asked how they would prove that $\sqrt 2 + \sqrt 5 + \sqrt 7$ is an irrational number. I've read the answers to that thread and I couldn't really "apply"/use them in my situation.
Exercise added for reference. Post was edited because it was referring to a particular case (where n was assigned a value such as 6).
I'm trying to reopen the topic because I am more curious as to how you'd solve this type of exercise in the general form (like in the image).