Final answer:
The Spanish established their rule in the Americas primarily through military conquest, using the encomienda system to reward loyalty with control over land and local labor. Their colonial administration was marked by rigid control and extraction of resources. In comparison, other European powers like the French focused on specific goals such as trading for furs.
Step-by-step explanation:
The Spanish rule over the Americas was established through military conquest. Notable figures such as Hernán Cortés, who conquered the Aztecs, played a key role in developing the Spanish empire in the New World. The Spanish Crown granted land and its inhabitants to loyal soldiers and encomenderos, leveraging their loyalty to both the Crown and the Church. Establishing colonial administration and a bureaucracy, the Spanish enforced rigid control and sent treasures and profits back to Spain. Their methods of control included military dominance, cultural assimilation, and economic exploitation, particularly through the encomienda system, which rewarded conquerors with the labor of local indigenous peoples.
In terms of other goals of colonization, such as those of the French, their main goal was trading, especially for furs, as this brought about significant profits and economic growth for France. Each European power had distinct objectives and strategies for colonization, and these heterogeneous approaches shaped the cultural and political landscapes of the Americas.