What is the difference between attenuate and extenuate in terms of their usage and connotation?
Generally, to attenuate is to dilute, and to extenuate is to excuse. While one sense of extenuate is "To make thin or slender; to draw out so as to lessen the thickness," the sense "To lessen; to palliate; to lessen or weaken the force of; to diminish the conception of, as crime, guilt, faults, ills, accusations, etc." is rather more commonly used.
 From wikipedia: "Attenuation is the gradual loss in intensity of any kind of flux through a medium." From wiktionary: "To reduce in size, force, value, amount, or degree; ... To weaken; ... To rarefy."
(I see that instead of "rarefy" being an alternative spelling of "rarify", wiktionary says it's the other way around.)
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