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With Explanation

This is an another confusing grammar. It is about 어떡해 vs 어떻게.

Like 왠 and 웬, both of them have same sounds. When ㄱ and ㅎ meet, it become ㅋ, so 어떡해 sounds like [어떠캐] and 어떻게 sounds like [어떠케]. Although there is a difference between ㅐ and ㅔ, it is really negligible.

Let's see these 2 expressions in more detail.

어떡하다

어떡하다 is shortened form of 어떻게 하다(doing something in some way). ㄱ in 게 went under 떠 and original ㅎ is eliminated because of ㄱ. That made the verb 어떡하다.

As 어떻게 하다 is used when we want to say "what should I do?" phrases. When you don't know what to do something, then you can use both 어떻게 하다 and 어떡하다.

우리 이제 어떡해(어떻게 해)?
What should we do now?

그렇게 말해 놓으면 형 보고 어떡하라는(어떻게 하라는) 거야?
If you told her something like that, what should I do?

우리가 어떡하면(어떻게 하면) 상대를 이길 수 있을까?
What should we do to defeat the opponent?

어떡하지? 순서가 뒤바뀌었는데, 바꿀 시간이 없어.
What should I do? The order is reversed, but I don't have time to change it.

어떻게

If you want to say what should I make/decorate or how should I study/teach/know rather than just simple do(하다), then use 어떻게.

블로그 어떻게 꾸미는지 알아?
Do you know how to decorate blogs?

한국어를 잘 하려면 어떻게 공부해야 하나요?
How should I study to speak better Korean?

그걸 어떻게 아셨어요?
How did you know that?

김치는 어떻게 만들어?
Can you tell me how to make Kimchi?