He always relies on Cody to do his homework and school projects as Cody is clearly the more intelligent of the two.
Zack is also a very good strategist, since the first episode he is shown to be able to make quite good plans that usually succeed or would succeed if an unseen second or third party didn't intervene. Martin", Zack changed history when he learned of the tampered compass.
Olsen doesn't give up, and attempts to merge the twins again. In "Goin" Bananas", he implicitly states that he would try harder; however, because of his brother's over-achieving personality, he explains that nothing he ever does is good enough.
Bailey ropes him using her lasso, rendering his plans futile. In line with this idea, despite Zack's obvious moments of intelligence, he can be short-sighted and lacking in his thoughts, making him appear dumb to others.
He traveled the nation with Cody and his mother Carey before finally settling into the Tipton Hotel as seen in the first episode of the series, "Hotel Hangout".