The purpose of the French expeditions was to establish a

Establishing themselves in Illinois and at Mobile, New Orleans, and Arkansas Post, French explorers traveled further into the interior of North America, opening relations with various tribes and also seeking a route to New Mexico to open trade with its Spanish inhabitants.

Explanation:

Background. The French first came to the New World as travelers, seeking a route to the Pacific Ocean and wealth. Major French exploration of North America began under the rule of Francis I, King of France.