Vietnamese culture has existed for ages.
Is the use of "for ages" correct in this sentence? In OALD, "ages" is an informal word. If so, I think the phrase "for ages" should not be used in this sentence. Please explain it to me.
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Yes, that is correct. It's using the following definition of ages:
4. ages Informal An extended period of time: left ages ago.
It is equivalent to:
Vietnamese culture has existed for a very long time.
It doesn't say exactly how long they mean; in the context of a culture, I'd guess at least 1,000 years. In other contexts, it could mean almost any length of time.
The other two answers are correct, but I wanted to specifically address the "formal" part.
"Vietnamese culture has existed for ages" would be fine in nearly all contexts. It would be too informal for, say, a scholarly history of Southeastern Asia, where you would want to be more precise: "Vietnamese culture has existed for XXXX years" or perhaps "Vietnamese culture has existed since at least XXXX".