When you buy vegetables at a farmers market, often you will get root vegetables (e.g. potatoes, carrots) that are unwashed and with its dirt still clinging on them. I heard somewhere that this will prolong its shelf life. Is this true? If so, how?
Not really, the dirt's there because of:
Washing to remove dirt won't shorten the shelf life, but mechanically removing dirt (as in with a hard brush) might in some cases as it could remove or puncture skins or peels.