If the gum is not smashed deep into the carpet pile, using your finger nails pinch – lift and snip with scissors.
If the gum is deep in the carpet pile:
Step #1: Put a cube of ice on top of the gum and freeze it. Now take the heavy end of a butter knife handle and hit the gum and it should shatter. You may need to do this several times to chip away it. Be patient it will eventually work.