For example:
老師: 你多大了?
學生: 回先生的話, 學生今年八歲。
Or 學生: 回老師的話, 學生今年八歲。
It sounds very polite to me but I do not hear it often, even in the Chinese culture, where teachers are generally respected.
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Sign up to join this communitystarting a reply with 回 is too subservient in modern Chinese conversation.
You only hear people use "回 + title" in period drama nowadays
In olden time, people would use "回 + title" to replay to authority
e.g. 回皇上, 回大人
They didn't even normally talk like that to their father or teacher (they are in higher status and position, but they are also close to you, therefore, no need for subservient phrase like that)
回先生的話, 學生今年八歲 imply the student himself acknowledges he is not allowed to speak in front of the authority (in this case, the teacher) unless spoken to or with permission to speak. It is not the society generally think in modern time.
At least this kind of phrases not usually happened in Taiwan. I feel that it is redundant, strange and not natural. I think 我今年八歲 is ok to reply.
Putting 回 in front of answering phrases may be outdated in most part of Chinese conversation nowadays. I believe that even seniors will find it strange when answered in this way.
It is only mainly heard in TV drama nowadays. Usually when it's the servant replying to the King or Queen after given permission to speak.