What is the difference between "u'Shemartem" and "va'Asisem"?
Rashi: The former refers to Mishnah (learning); the latter, to carrying out what one learned.
Tzror ha'Mor Eikev (DH ve'Hayah): "u'Shemartem" corresponds to the La'avim; "va'Asisem", to the Mitzvos Asei.
Targum Onkelos and Targum Yonasan: They refer to guarding (keeping in mind) the Torah and doing what it says.
What is "Hi" referring to?
Rashi: It is referring to the learning and the carrying out mentioned earlier in the Pasuk. 1
Targum Onkelos and Yonasan: It is referring to the Torah, which "u'Shemartem va'Asisem" themselves are referring to.
Rashi: And it is as if it says 'through the learning mentioned earlier, you will be considered wise men
What are the connotations of "ki Hi Chochmaschem ... le'Einei ha'Amim"?
Rashi: It is through the study of Torah and its observance that we are considered wise and understanding in the eyes of the nations.
Seforno: It is by means of the Torah that we are able to answer the Apikorsim with logical arguments. 1
Shabbos 75a: It refers to the calculation of Tekufos (solstices and equinoxes) and Mazalos (the times and positions of the constellations). 2
Why does the Torah write "le'Einei ha'Amim", and not 'Neged ha'Amim'?
Avodah Zarah, 4b: To teach us that in Olam ha'Ba, all the Mitzvos that Yisrael kept will come and strike the nations of the world (they will be embarrassed) - that they spent all their time on matters of no consequence. 1
See Torah Temimah, note 8.
Why is it important to be considered wise in the eyes of the nations?
Seforno: Because, as the Torah goes on to explain, Hashem is close to us inasmuch as He answers us when we call to Him, a clear proof that He chose us from all the nations, and it would be a Chilul Hashem if the nations would consider us foolish.
Oznayim la'Torah: Because, in addition to fulfilling the Mitzvos, it creates a Kidush Hashem when the Nochrim see us doing so. 1
See Oznayim la'Torah.