Final answer:
President Theodore Roosevelt's action that illustrates 'speaking softly' was negotiating treaties and agreements, emphasizing diplomacy over military strength.
Step-by-step explanation:
The action that demonstrates how President Theodore Roosevelt could "speak softly" as opposed to using a "big stick" would be negotiating treaties and agreements. This approach is aligned with the aspect of his foreign policy that emphasized diplomacy and negotiating skills over outright military intervention. Roosevelt's use of the phrase, "speak softly and carry a big stick", highlighted his strategy of using the United States' growing international power to influence other countries while avoiding direct conflict when possible. An example of this method is seen in Roosevelt's role in negotiating peace treaties, such as the one that earned him the Nobel Peace Prize for helping end the Russo-Japanese War.