KGYSAK: 안 vs 않

You've been there. Constructing a sentence and when pressed to spell "not" as in "안 했어요" you pause for a moment and wonder... should you spell it "않 했어요" instead? Which one is correct? You've seen both, haven't you?

Sort of. They have the same basic meaning but a different usage.
'안'한다 (don't) - 하지 '않'는다 (don't)

안 할게요
I won't

안 합니다
I don't

안 했어요
I didn't

안 잤어요
I didn't sleep

안 갈거야
I'm not going

아직 안 왔어요
She/he hasn't come yet.

안 먹어?
You're not eating?

않다 also indicates negation but in a different sense

제가 하지 않았습니다
I didn't do that.

그렇게 하지 않나요?
Don't you do it like that?

다시는 실수하지 않겠습니다.
I won't make a mistake again.

먼저 가지 않을게요.
I won't go first.

쉽지 않아요
It's not easy

So in the first example, it is indeed 안 했어요 but do keep in mind that 않다 is its own thing and that it has a place. When I'm questioning it, I think to myself of the placement of 안 or 않. If it's in front of an action verb like 하다 or 가다 I use 안 as in: 공부 안 했다 or 학교 안 가. Where it gets fuzzy is situations like "aren't you tired?" where the message can be conveyed two different ways:

안 피곤 해요?

just watch your 안s and you'll be fine. More examples:

안 먹어요
안 자요
안 예뻐요
안 추워요

먹지 않아요
자지 않아요
예쁘지 않아요
춥지 않아요

Thus concludes another Korean Grammar You Should Already Know.