Final answer:
America's empire clashed with the nation's democratic values through colonization and imperialism, challenging traditional principles of liberty. The expansion raised constitutional questions and internal debates, illustrating the problematic nature of empire building from a democratic standpoint.
Step-by-step explanation:
The ways in which America's empire challenged Americans' belief in liberty and democracy include colonization and imperialism, which in practical terms meant exercising control over other peoples and territories. This control at times contradicted the principles of liberty and equality that form the cornerstone of American democratic ideals.
The belief in American exceptionalism and the duty to spread democratic values underpinned the imperialistic expansions, but the execution of such policies led to internal debates and concerns about empire building running counter to American democratic principles.