I am learning the Old Babylonian language and just stumbled over the word/form pāqidūtum. It seems to be a third person male stativ singular + u + the female ending tum of the verb paqadum (to care for), but that does not make any sense to me. Does anybody have an idea?
pāqidūtu(m) is the nominative plural of pāqidu(m), the active participle of the verb paqādu “to entrust etc.”
(In theory it could also be an abstract noun in -ūtu, as suggested in the other reply, but such a word, if it existed, would have merited a separate entry in the lexica. Also, the fact that the author does not introduce this suffix until a later lesson indicates that he would not expect students to know it at this point.)
The word pāqidūtum is not listed in the CAD, which suggests that it's likely not historically attested. But the word pāqidu(m) = "provider, overseer, caretaker", also included in the exercise you cited, is listed on CAD volume 12 (P) page 137.
One page 35 of the lecture notes you linked, you will also find the abstractifying suffix -ūtum, used to derive abstract terms for "the state of being something" (and sometimes collectives like "all somethings") from concrete nouns, such as šarrūtum = "kingship" from šarrum = "king" or awīlūtum = "humanity, mankind" from awīlum = "(free) man, person".
Thus, by analogy, pāqidūtum can be fairly regularly and transparently analyzed as "providership" or "the role of a pāqidum".