What is an example of the use by the United States of its military power after World War II?
The above answer is not particularly informative, so hopefully this addition will provide greater context and a more useful example.
The United States has used military force hundreds of times, ranging from small scale military actions in remote corners of the world to massive deployments of military personnel and equipment to respond to invasions of friendly or allied nations in Europe, Asia and the Middle East. The armed forces have been used to liberate countries occupied by hostile regimes, to assist foreign governments in responding to natural disasters, to remove hostile governments, such as occurred in Iraq in 2003, and to eliminate havens exploited by terrorist organizations such as has occurred in Afghanistan and Somalia. Some of these military actions have enjoyed popular support in the United States, and some have not. Just since the end of the Second World War, American presidents have dispatched troops and ships to the Balkans, Southeast and Northern Asia, Central America, the Middle East, small island nations in the Caribbean like the Dominican Republic and Grenada, and to the polar ice caps to assert or reaffirm U.S. sovereignty over disputed territories. Some of these military actions involved Cold War considerations -- and the example of the Korean War provided in the earlier answer certainly qualifies -- and some did not. The Cold War's end in 1989, followed by the dissolution of the Soviet Union two years later, has been followed by many non-Cold War-related military actions, including the aforementioned wars in Iraq and Afghanistan, military interventions intended to end fighting in Bosnia-Herzegovina and Kosovo in the former Yugoslavia, the effort at facilitating the delivery of emergency food supplies in the disasterous military intervention in Somalia in 1993, military intervention in Haiti in 1995, and many others.
While the war in Iraq initiated in 2003 has been mentioned, a good example of a post-World War II use of the American military involved Operation Desert Storm, the 1991 military campaign that forced Iraqi troops from the Persian Gulf nation of Kuwait, which Iraq had invaded in August 1990. Iraq's invasion, potentially the result of diplomatic incompetence on the part of the U.S. officials involved, was characterized by very brutal actions of the part of the Iraqi troops in subjugating their Kuwaiti victims. Following months of diplomatic efforts at convincing Iraqi President Saddam Hussein to remove his troops from Kuwait, including the use of economic sanctions in an effort at compelling Iraqi compliance with United Nations resolutions demanding such withdrawal, then-President George H.W. Bush ordered U.S. troops to begin the liberation of Kuwait. Six weeks after the order was given, the last Iraqi troops had been forced out of Kuwait by U.S. and allied troops. Thus began the decade-long stalemate over whether Iraq continued to possess nuclear, chemical or biological weapons which were prohibited to it under the terms by which the U.S. agreed to cease its military attacks on Iraqi forces. The culmination over that stalemate was the invasion of Iraq ordered by President George W. Bush in March 2003, following which intense searches for the aforementioned nuclear, chemical and biological weapons revealed that all such weapons had been secretly destroyed by the Iraqi government several years prior, thus indicating that the occcupation of Iraq had been initiated under mistaken pretenses.
In the time from the end of World War II to the early 1990s, the US typically used its military power for Cold War purposes. That is, it used its military power to try to prevent the spread of communism. This military power was sometimes able to achieve the country’s purposes but it was not always able to do so.
One time when US military power was able to, for the most part, achieve what the country wanted was in the Korean War. This war began in 1950 and ended in 1953. The Korean War began when North Korea, which was communist, invaded South Korea, which was not. North Korea wanted to reunify Korea under communist rule. The US led a UN coalition to eject North Korea from the south. This was a conventional war fought between large armies. The military power that the US had at its command was able to defeat North Korea fairly quickly and push it out of the South. The war did, however, drag on when the US overreached and brought China into the war. China’s huge amounts of manpower made it impossible for the US to hold North Korea and the war ended with the boundaries between North and South Korea in essentially the same place they had been when the war began.