In an extreme or severe manner; to a very great degree or extent.
From Greek 'drastikos' meaning 'effective, active,' from 'dran' meaning 'to do, act.' Originally used in medical contexts to describe powerful medicines or treatments. The sense of 'extreme' or 'severe' developed from the idea of medicine that acts powerfully on the body.
The word maintains its medical DNA in everyday usage - when we say something changed 'drastically,' we're unconsciously invoking the image of powerful medicine that dramatically alters the body's state. It's stronger than 'significantly' but less violent than 'catastrophically,' occupying a perfect middle ground for describing major changes.
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