Easiest way to cut into a concrete slab??

Rent a gas powered wet concrete saw. Your patio is probably not more than 4" thick. The saw will cut all the way through. Your slab may be a little thicker around the perimeter Say 6". Then take a sledge hammer and brake the cut strip in 1 place. You should be able to pry the rest out with a breaker bar. The wet saw is much better for depth control easier to follow a line & eliminates dust.