instrumental singular + api
Analysis. kukkuṭapattena (instrumental of kukkuṭapatta, "chicken feather") + api ("even, also"); the compound kukkuṭapatta itself breaks into kukkuṭa ("rooster") + patta ("wing, feather"); api is an enclitic particle meaning "even" or "also".
Etymology. kukkuṭa is onomatopoetic (the cry of a rooster; compare Sanskrit kukkuṭa and modern Hindi kukṛ). patta derives from √pat- ("to fly, fall"), cognate with English "fathom" and Latin patēre ("to lie open, extend"). The whole phrase emphasizes "even a chicken feather" in contexts of non-attachment or minimal value.
Meanings from the Digital Pāḷi Dictionary; morphology and etymology by Claude.
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