I deeply regret that I did not think of testing this when I was in school. There are infinite answers to any “solve for x” problem, and I now realized the many missed opportunities to be a precocious wise ass.
The repeating decimal is an interesting case here. By comparison, if the question is (2-1=x), then (x=2^0) is true, but I suspect the examiner would say that it’s not in its most “reduced” form, and that seems legitimate since there’s an unresolved operation baked in. But I don’t think that objection works as well for (0.\dot0): it’s not really any less “reduced” than (1), at least not as obviously as an answer that contains an operation.
Ultimately, “is it correct” and “would it be considered correct” are different questions. It is correct, but I suspect many teachers wouldn’t accepted it. Lots of teachers punish wiseassery, even when it obeys the letter of the law, and appealing to mathematical trivia to provide a technically correct but counterintuitive and less-clear answer would be read as being a wise ass.
I haven’t read much at the intersection of math and philosophy of language, but my impression is that popular conceptions give it more meaning than it needs. I think it’s possible to build math as a purely formal language, with no real-world analogues to the transformations and relationships necessary to specify how its objects are interconnected. Define rules about how we can manipulate symbols, and then use the rules to show that two symbols satisfy the relationship “=”.
But that’s beyond me, and I’m only about 80% sure it’s true.