How can I know, precisely, when to differentiate the sounds of the letter a, like in: apple and vault?
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This is an extremely broad question, actually, and I doubt there can be a single, definite, comprehensive answer. I will try to provide a few quick-and-dirty rules of thumb, but be aware that you will probably find exceptions to every single one of them.
Also, as Colin Fine points out, note that things are not pronounced the same everywhere. For example, can't can be pronounced as /kaːnt/, /kɑːnt/, /kænt/, and /keənt/, depending on what variety of English we are talking about. As a second example, the vowels in Mary, marry, and merry sound identical in certain dialects but not in others.
All that being said, on to the rules of thumb.
First, the basics. Here are the Wiktionary usage notes for a:
Now on to details, which I will be more adapting than quoting from the essay "Hou tu pranownse Inglish" by Mark Rosenfelder. Note that the order of the following rules of thumb is important. As the author puts it, "to pronounce a word, you go down the list of rules, seeing if each one in turn applies, and applying it if it does".
Again, these "rules" are only approximations, so take them with a grain of salt (also, I hope I haven't thrown in a few typos or copy-paste mistakes).
For further reading, see these questions:
Unfortunately English orthography is difficult. There are patterns (or rules), but there are sometimes conflicting patterns, and also many exceptions.
There are also further complications where pairs of words are distinguished in some dialects but not others: for example in my (UK) accent "ant" and "aunt" sound quite different (but "aunt" does not have the same vowel sound as "vault", and "aren't" sounds the same as "aunt") but in some American accents "ant" and "aunt" sound the same.
So it is difficult to give you a clear answer. "Au" is certainly a different environment from "a", and in most cases will be pronounced differently.