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I can't even go one day without this dictionary.
Literal
I [topic-は] 1 day even this dictionary without get-by thing [subject-は] cannot-do.
さえ intensifies — 'even [the minimal case].' 辞書無しで済ます literally 'to make do without a dictionary.' 無しで is a fixed adverbial 'without,' and 済ます ('to get by, to make do') takes it as a complement. The pattern [verb]+事は出来ない is a formal/written 'cannot [verb],' equivalent to the potential form but sounds more considered.