What are some good answers to questions e.g. "why do we need to study square roots"?
Of course the answer depends highly on who is asking. For the scope of my question, I have a student in mind, who is - given their interests for further education - likely never going to use square roots in adult life, but still has to learn them. (Of course my question is not about square roots specifically. It can be any topic which, realistically, will not be used "in real life" by the student.)
I have some answers, but I feel that they all fall short of being a satisfactory answer:
"Because you have to know it for your grades, so later you will have more choice of what you want to do in life" - this is my honest answer, but I fear it might be a source of resentment for the student against the education system, which I don't want.
Bring some forced example from "real life", e.g., "John has a son whose age is the square root of his age. John is 49 years old. How old is his son?" - I want to avoid these at any cost, as any student will see through these and will be enforced in his view, that "math is nonsense". Even if the example is better and I think more relevant "in real life", if it fails to resonate deeply with the student, then it will backfire as an example.
"It teaches you to think abstractly" - I am deeply skeptical of some generic problem solving ability. I think we are good at solving problems that we practice solving, and it's not obvious how one transfers to another problem. And even if generic problem solving activity is a thing, it is hard to see why practicing square roots is the best thing to improve it, instead of something else that the student is more interested in. I think, hearing this answer, most students will nod along and still think to themselves that math is nonsense.
"Distract from the question". The "why learn this" question usually arises from a thought process, that is just distracting the student. He finds square roots hard to understand, so he is looking for an excuse to think about something else. He never seems to ask the "why do this" question for activities that he otherwise just enjoys, e.g., reading a novel, playing music, drawing etc. and if asked "why", he will rationalize why it is a good thing to do. So anything that distracts from the "why question" and manages to bring focus back to thinking about the problem itself is good. (If you have any concrete strategies for this, I appreciate if you share it.) However: I still think, that the "why learn this" question is important and I would not like to ignore it completely.
I am an applied mathematician myself, so I can bring real life examples where math (beyond basic arithmetic) is useful. But the student will likely not have the knowledge to fully appreciate it, so while he might be impressed momentarily, I don't think I can expect any consistent success. It certainly makes sense to try connecting the applications of math with school curriculum every once in a while, but honestly these are often very far away from each other.
Is there any good strategy to answer to the "why learn this" question, that I missed?
Did I make a conclusion above, which is absolutely wrong?
If possible, I am looking for answers which are based on at least some empirical evidence, but I am also interested in concrete examples where something has worked really well.
EDIT: thanks for all the contributions. Some further clarification about my question:
I look for direct answers and not analogies. Math is not sports, art, or music. To me, such answers are essentially distraction (see point 4 above).
Several answers are in the direction of "it's part of general knowledge". I see this as appeal to authority and as such, the nicer version of appealing to grades (point 1 above).
Some answers are quite specific, e.g., lots of suggestions are about finance (which btw I personally find really boring). Some topics in math connect better with finance than others and having one "go-to example" isn't a really useful mindset when dealing with a concrete student and a concrete topic.