Because we're arrogant son's a *******. :D

Post WW II. Ours was the only economy left standing.

The United States has a huge population, so it's size alone makes it powerful. It made huge profits out of two World Wars, leading to it being able to afford what no other country could afford in the immediate post-war years. Also, and probably just as important, it has Hollywood, which has done a superb job of convincing virtually all Americans and a huge proportion of the rest of the world that its fantasy is actually reality.

Nearing the end of WWII, the US economy (as stated by Zazzy in another answer) was the only one left standing. With the formation of what would eventually become the United Nations and the International Monetary Fund (IMF) the US was looked upon to lead the way in world wide Economic recovery. Eventually the US$ would replace the gold standard as the base of finances in many countries and to this day is the standard that all other ecnonomies are measured against. The US is still a world power and will continue to be so because of big business and its overall worth to the rest of the world, however the European Union is gaining stregnth with each passing day and now the African Union is taking hold as well, and all of this is happening as the US$ is weakening and the Euro is gaining stregnth.

The question was about 'a world power' not 'the world power'. The U.S. gained it's dominant role during the 20's and securing it in the 1940's. But it was 'a world power', ie, one of the most powerful countries in the world, already by 1860. The U.S. had begun becoming an imperial power as soon as it fully settled its own borders at that time. It was founded during the birth of the industrial revolution, and immediately became a center of innovation and modernity.

When we proved we had no problem dropping atomic bombs on cities and killing tens of thousands of innocent people.

Well, the founding fathers didn't sit down in a tavern in Boston one day and say, "Hey, Why don't we make a world power?". It just kind of...happened. It happened because of the American people's unity, ethic, creativity, ambition and hard work. People being free to think for themselves and be accountable for their actions and recognizing that all people have the right to live life as they see fit. That's what the revolution was all about anyway. WWII is more of a factor in bringing the US out of the depression than it is in establishing it as a world power. The US began to emerge as a such in about 1804 allying itself with the Kingdom of the two Sicilies against the Barbary Pirates, remember the Marines hymn? , ...to the shores of Tripoli." The US defeat in 1812, of the same British army that defeated Napoleon brought some attention. Then in the 1820's the Monroe Doctrine set the US up as protector of the Americas and establish it's right to resist Europien colonization in the Americas. US elimination of Europien involvement in the Americas played a part i.e. the Louisiana Purchase, Texas Revolutiion, Mexican War though the two latter were more factors in defining the shores of the US there was still an international effect. As its Navy grew and visited other countries people began to recognise the viability United States. China in the 1850's as well as relations with Japan about that same time. Finally in 1898 when the USS Maine was sunk in Havana Harbor the US went to war with Spain and won the Philippines, Puerto Rico, Hawaii, Guam and the right to establish a naval base in Cuba, at Guantanamo Bay, a place we all know and love. In about 1902 US involvement allowed construction of the Panama Canal...so and so forth, WWI, then after WWII the US involvement in rebuilding Germany and Japan. It seems that we have always had plenty and were willing to share it to help somebody in need. Through all of the conflicts in the last two hundred years because of a strong desire to remain neutral in a variety of conflicts it was ultimately forced to involve itself; I believe that it would be very difficult to legitimately paint the US as the worlds aggressor to establlish itself as a world power.

The initial intent was to create confusion. Both bombs did exactly that; in different ways. Incidently, I either saw a documentary or read that the White House stewards talked among themselve about how Truman drank one night, they figured it out later. That tells me that the result of the decision was known before the decision was made. At any rate the atomic bomb had little to do with establishing the US as a world power.