1. First, cut up all the vegetables and chicken.

2. First, cut all the vegetables and chicken.
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Why is 'up' used in Sentence 1? Can't we omit 'up' as in Sentence 2?

Yes, I think you can delete the word "up". I would use it, because that's what we say, but to cut the vegetables is just as accurate and means the same, really. The only difference might be that cutting something up implies small pieces, while cutting it is vaguer.