My new VHF antenna got whacked by a tree branch. It was a clean break about 3 inches from the end. The inner core cable which carries the signal is hanging out by 3 inches. If I were to cut this off how much would the 3 inches from an 8 foot antenna effect its operation?

Don"t cut it off! If you do, you might as well toss the antenna. The way these antennas are designed, and the way they are tuned makes them somewhat narrow banded. cutting off 3" will send it so far off frequency it will be useless. The actual antenna section consists of sections of coax cable cut for the operating frequency, that upper piece of wire that you're seeing, and a tuned "stub". The whole assembly is not that big, and actually sits high up inside the fiberglass tube, with the coax feed line running up the tube to it. If you can open the bottom of the antenna, you should be able to pull the whole assembly back 3" to bring that wire back into the tube with no ill effects. My antenna was damaged when I bought my boat, and that's exactly what I did. Actually, I went a bit further and removed the antenna assembly and hand tuned it with an antenna analyzer, and a signal generator (But I digress). Just pull it back and cap off the end. You should be fine.

OK, sort of thought that but the redneck button was close by the scissors. I will see if I can pull it down from the bottom.

"Redneck Button" I like that.