Final answer:
The Privileges and Immunities Clause of Article IV Section 2 of the U.S. Constitution guarantees that any American can travel or live in any state and be entitled to the same rights as residents of that state.
Step-by-step explanation:
The clause that ensures any American may travel or live in any state without being treated differently is the Privileges and Immunities Clause found in Article IV Section 2 of the U.S. Constitution. This clause provides protection to citizens by entitling them to the same rights and privileges as the residents of the state they are visiting or residing in. While there are nuances, such as out-of-state tuition being higher for non-residents at state universities, the clause is a fundamental part of ensuring equality for citizens as they move across state lines.