Hi there, I am attempting to replace the slip joint nut under my kitchen sink. The old one essentially disintegrated and broke off. I bought the replacement piece however I realized that I can't get the nut onto my copper pipe because the dishwasher waste nib is in the way. How can I replace the nut then?

I will guess you are talking about a tailpiece nut. You need to replace the whole tail piece as the DW connection is integral to the piece. They are less than $5 at HD.