>>Add a new step 1. Remove the quarter round and baseboards
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>Neither of those to worry about.
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>>and cut the bottoms off the door and closet molding (assuming you've already pulled the carpet) to the height of the new flooring.
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>Is there any power tool that can do this? Or do you have to get handy with a hand saw or even a hack saw? It seems like the cuts will be too low for a power jigsaw or circular saw.

You can do it with a power saw, but use a very fine toothed blade.