'independently' originates from Latin, specifically the word 'independens,' where 'in-' meant 'not' and 'dependere' meant 'to hang down.' 'Operate' originates from Latin, specifically the word 'operari,' meaning 'to work.'

'independens' transformed into the Old French word 'independent,' and eventually became the modern English word 'independent.' 'Operari' transformed into the Old French word 'operer,' and eventually became the modern English word 'operate.'

Initially, 'independent' meant 'not dependent,' and this meaning has largely remained the same in modern usage. 'Operate' initially meant 'to work,' and this meaning has largely remained the same in modern usage.