Final answer:
The Spanish utilized the encomienda system to control Native Americans by exploiting their labor and attempting to convert them to Christianity, which later evolved into the repartimiento system as a means of maintaining control.
Step-by-step explanation:
One method the Spanish used to gain control over the Native Americans was the imposition of the encomienda system. This system involved Spanish authorities assigning native workers to mine and plantation owners with the expectation that these owners would defend the colony and indoctrinate the workers with Christianity.
However, in practice, the encomienda system exploited native workers, compelling them to labor under often harsh conditions with little regard for their welfare or beliefs. When the encomienda system was seen to be unsustainable, it was replaced by the repartimiento system, which continued to enforce labor obligations on Native towns for the benefit of Spanish overlords, thereby solidifying control through labor and cultural dominance, including forced conversion to Catholicism.