Final answer:
European powers established colonies in the Americas to gain wealth, fuel industrial growth, and dominate rival powers.
Step-by-step explanation:
European powers sought to establish colonies in the Americas for several reasons. Firstly, they believed that the Americas were rich in resources and treasures, which would bring wealth and glory to their respective nations. Secondly, colonies played a crucial role in the mercantilist system, providing raw materials for industrial growth and creating a ready market for manufactured goods. Lastly, European nations aimed to dominate rival powers by establishing colonies and exploiting their resources.