What is the meaning of "Atah Yadcha"?
Radak: It means b'Yadcha; the prefix Beis is omitted, like "ha'Nimtza Beis Hashem" (Melachim II, 12:11). You expelled nations with Your hand. It was not via Yisrael; they did not need to fight.
Malbim: You dispossessed the nations to plant Yisrael in place of them. This was not via intermediaries or nature, only via Your hand. (a) You dispossessed them for the good of Yisrael, to plant them in the land. (b) It was due to the nations' evil.
Why does it say "va'Tita'em"?
Radak: They will stay there forever, as if they are planted in the land. The same applies to "Tevi'emo v'Sita'emo" (Shemos 15:17).
What evil did Hashem do to nations?
Rashi: He expelled the seven Kena'ani nations from in front of us. He dispossessed them, and planted us in the land in place of them. Radak - the Lamed in l'Umim is a prefix; we find the root Umim without Lamed, e.g. "Shabechuhu Kol ha'Umim" (117:1).
Malbim: [He expelled them] due to their evil - "b'Rish'as ha'Goyim ha'Eleh Hashem Elokecha Morisham Mipanecha u'Lema'an Hakim Es ha'Davar Asher Nishba" (Devarim 9:5).
What is the meaning of "va'Teshalchem"?
Radak #1: You sent [the nations from in front of Yisrael].
Radak #2: You killed them; this is like "b'Shelach Ya'avoru" (Iyov 36:12).
