答 is bamboo ⺮ over 合 'to fit together'. Replies once traveled on bamboo slips, and a good answer is one that fits the question — the character weds the medium to the logic.
回答 is to answer, 答案 the answer key every student hunts for, 答应 to agree or promise. 答 is the rhythm of the classroom: 问 asks, 答 answers, and exams are 答卷 — answering the paper.
Bamboo ⺮ slip that FITS 合 the question — an ANSWER.
答应 'answer-respond' means to promise — in Chinese, merely answering someone edges toward commitment, which is why silence is sometimes the safer reply.