When he was cast out (εκτεθεντος αυτου). Genitive absolute with first aorist passive participle of εκτιθημ.
Took up (ανειλατο). Second aorist middle indicative (with first aorist vowel α instead of ε as often in the Koine) of αναιρεω, common in the N.T. in the sense of take up and make away with, to kill as in verse Acts 7:28, but here only in the N.T. in the original sense of taking up from the ground and with the middle voice (for oneself). Quoted here from Exodus 2:5. The word was used of old for picking up exposed children as here. Vincent quotes Aristophanes (Clouds, 531): "I exposed (the child), and some other women, having taken it, adopted (ανειλετο) it." Vulgate has sustulit. "Adopted" is the idea here. "After the birth of a child the father took it up to his bosom, if he meant to rear it; otherwise it was doomed to perish" (Hackett).
Nourished him for her own son (ανεθρεψατο αυτον εαυτη εις υιον). Literally, "she nursed him up for herself (εαυτη besides middle voice) as a son." This use of εις=as occurs in the old Greek, but is very common in the LXX as a translation of the Hebrew le. The tradition is that she designed Moses for the throne as the Pharaoh had no son (Josephus, Ant. ii. 9, 7).