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I found this sentence in a book:
家里的空调坏了，咱们另外再买一个吧。
I don't understand why I have to write 再 inside this sentence. What is the use of 再 in the sentence and what does it do to complete the meaning of the sentence? Doesn't 再 mean "again"? Why is there a 另外 in front of the 再, doesn't it mean "other than that then we will buy another one"
另外 - In addition; other (adv)
再 - again (adv)
买 - buy (v)
Both 另外 and 再 are adverbs for the verb 买 in this sentence
Why both?
Because the sentence 另外买一个 could mean either 'buy an additional one' (now you have two air conditioners) or 'buy another one' (to replace the broken one)
If you say 再买一个, it could mean either 'buy an additional one' (now you have two air conditioners) or 'buy one again' (to replace the broken one)
Since the sentence already stated 家里的空调坏了, there's no reason to think you will have two air conditioners, therefore 另外买一个吧 would be good enough. Adding 再(again) avoids confusion and confirms the fact that the new one is a replacement. (we bought one before and now we have to buy one again - not the same one, another one)
"家里的空调坏了，再买一个吧"? is grammatical, but it would sound like you replace air conditioner often
家里的空调坏了，咱们另外再买一个吧。
家里的空调坏了，咱们再买另外一个吧。
家里的空调坏了，咱们另外买一个吧。
家里的空调坏了，咱们再买一个吧。
All these sentences are correct and have the same meaning.
If you already have one or more (Or used to have), then you want to get another one or more, use 再 or 另外 or 另外再 are all right.
e.g.
我剛失業了，要再找另一個工作了。(I just lost my job, I need to find a new job.)
我們買一台車吧！(we have one car now)
我們再買另外一台車吧！(we have two cars now)
不然另外再買兩台吧！(we have four cars now)
咱们（以前）买了一个空调，（现在）家里的（这个）空调坏了，咱们另外再买一个吧。
We have brought an air conditioning before, now we are going to buy again. If other person give you an air conditioning, you couldn't use 再 here. And you don't want to emphasize this thing or you have not this habit, you can delete it.
We have a fridge, now, We are going to have another one. So you can use 另外 here. But if you don't want to emphasize "we have a air conditioning" or you have not this habit, you can delete it.
家里的空调坏了，咱们买一个吧。
is same meaning.
'再' means again, '另外' means other. They are both adverbs for 'buy'. So the speaker is suggesting buying an air conditioner again, but this time you can not just repeat our previous order, the new one must be different from the previous one.
The purchase of air conditioners is going to happen again, but it will be a slightly different purchase.
