What is the difference between attenuate and extenuate in terms of their usage and connotation?
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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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