How powerful is the queen of England

Since you live in the USA, you are used to the "head of government" and the "head of state" being the same person. It is pretty common in the Western Hemisphere where most countries followed the lead of the USA,

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In Europe it is the working assumption that these should be two different people. Politics is dirty and messy and someone who is good at that might not be the best person to represent and personalize the state. In most of Europe a second person is elected, and is given the job of representing the state ,but without any real "power" over the ability of who to hire and fire, and over the military. Several countries have retained their traditional monarch. Norway voted for a monarch in 1905 after several centuries of not having a monarch because they were independent. Spain restored their monarch after a 40 year absence, because they liked the way he represents the country.

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Britain spent centuries fighting back and forth with their monarch. Finally about 150 years the relationship was simplified to "she reigns, but does not rule". The queen has a right to probe her ministers, and question them vigorously about the affairs of state. She represents the country abroad, and she has a large role in charities. She encourages the nation. However, she has reserve powers, which she can use if the country is in danger of collapsing.

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Although you are accustomed to Barack Obama making some power moves (increasing troop level in Afghanistan, etc.) a lot of what he does is not a power move, but one of encouragement. The century old term for this is he uses a "bully pulpit", meaning that he can encourage people to try and do things he thinks is important, but that he directly has not power. Think of Obama's visit to Ghana, his beer summit, his adress to students, etc.

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There is virtually no one in the UK that would like to replace the government with an AMerican style president (or to increase the power of the Prime Minister to be the same as the POTUS), but there is a small vocal minority that is in favor of adopting a European style republic.