This one thankfully is less complicated. Re- means again, or undoing, and often it’s just used as an intensive. It showed up in the thirteenth century (I think that’s the first date we’ve had for one of these) from the Old French and classical Latin versions of the prefix. Not much else is known about this one, although I’d like to point out that it’s also red- in front of vowels, so something redact is actually re- + act. Yet it’s not react. Just… just don’t think about it too hard.
A- the prefix is really three different prefixes, so we’re starting with confusing right off the bat The first of them is actually related to on, and it’s what’s in front of words like ahead and asleep—although it also can mean of, as in anew, and it can also also be short for ad- (to/toward) and ab- (away). The origin of this a- is unknown, but it is Germanic in some way. The second a- means away—much like the previous a-, and it comes from the classical Latin ab, away or from. The final a- prefix means not or without—like in amoral, you’re without morals. This one is actually Greek in origin, as a- meant not there, and it comes from the Proto Indo European ne-, not. And as I mentioned last week, that’s the origin word for un-. Well, one of the un-s.
Apo- means of, from, or away from—which means apologetic is literally away from (apo-) collection (-leg), and no, that makes no frigging sense, but whatever. This isn’t about that word. Apo- is from the Greek apo, from, which is from the Proto Indo European root apo-, off or away. Any time a word begins with apo-, it means away from something.
I don’t think this one is going to be much of a surprise. Anti- means opposed to or against, and is also just ant- in front of vowels. It’s from the Old French anti-, from the classical Latin anti-, from the Greek anti, and those all just mean anti. And it’s from the Proto Indo European… anti. Meaning against. Seriously, this word barely changed over thousands of years and through probably hundreds of languages.
The almost homophone for anti-, ante- means before or in front of and comes from the classical Latin ante, before. And that word is from… the Proto Indo European anti-. Yes, it’s from the same place as anti-. The only difference is that this one never went through Greek before it came to English.
Online Etymology Dictionary
University of Texas at San Antonio’s page on Proto Indo European language