You've got a great smile
You have a great smile
which one is correct or better to use when I am trying to compliment a lady, and why is that?
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Both are correct for "speaking" English. At least American English. Not sure about other types.
You've got a great smile.
Is technically incorrect in some circles but it is a very very common spoken and idiomatic usage. It's more informal.
You have a great smile.
Is generally more formal or in your case, could be taken as trying to be serious. So honestly it is one of those things that depends on the social situation.
There is a nicely detailed discussion of "you have" versus "you have got" on the English Language and Usage stack exchange here.
Your question, though, doesn't include "you have got" as an option . . . rather "you've got," which raises a distinct issue about using the contraction. Contractions generally sound less formal and more conversational. For the specific example you raise - "a great smile" - the contraction conveys a little informality which (to my ear) suits the compliment: you're saying something a little familiar, in a manner of speaking which is a little familiar. So either is correct but the contraction might be an especially good match for this sentence.