Answer: Crab grass is an annual grass and not deeply rooted the way you are describing, it also does not become evident until late spring. It is probable that you have some other weedy type of grass. Unfortunately, some of these are extremely difficult to control without chemicals, in particular an herbicide containing glyphosate applied according to the label instructions. To smother it, you would need to make sure the soil is moist, then cover it with either cardboard or five or six sheets of newpaper, then cover that with a heavy layer of mulch. Then you would need to wait for months until you are sure the grass is dead. Some grasses, however, particularly those with strong rhizomes, will survive this to come back and haunt you; tilling it breaks the rhizomes and creates a new plant from each little piece. The organic matter such as compost or peat moss should be added about two weeks or so before you plan to plant, otherwise it will be breaking down during the grass killing process and be of much less volume when you eventually plant.
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