i usually carry a folding camp shovel with me. not one of those crappy walmart ones that will bend and break on you the first time you have to do more than lift a scoop of sand–but a good army surplus one.
if i don’t happen to have that with me, i’ll try to improvise some kind of digging stick. hard woods are better than soft, naturally. sharpen the end with only one bevel. in other words, it should look like this:
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and not like this:
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if you bevel it from both directions, then you lose the strenght of the ring growth to push against, and you run a better chance of it splitting. to bevel, i will either cut the stick that way with my camp saw or grind it at a 45 degree angle on a stone or concrete surface.
shove your digging stick in the ground next to the plant with the beveled face away from the plant and then lever your stick away from the plant as well, so that the solid ring growth is taking the force of the earth and pushing toward the dandelion.
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(flower) (stick) lever the top of the stick this way -->
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(root) (bevel)
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your goal is not always to use the stick to push the whole root out of the ground. it depends on the soil. sometimes it’s better to just use the stick to losen the soil and then uproot you plant by grasping the stalk near the base and pulling it up–like “weeding” a garden.
with dandelions, the fiber strenght is pretty fragile. so i usually have to pull up the clot of ground the root is growing in and then use my fingers to break the dirt away from the root.
i lieu of a good, solid wooden stick, i have also used rebar and the tines of a turning fork (a gardening tool that looks like a pitchfork).
its the kind of thing you just have to experiment with. every plant will uproot differently. every soil will have different qualities. every digging stick will have different properties. but the principle remains the same.