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How can we understand which country is more powerful when we know the rate between their currencies?

Abbreviated Question: 
How can we understand which country is more powerful when we know the rate between their currencies?
Answer: 

The rates don’t reflect relative power. For example, 1 UK pound equals about 1.56 USD. That does not at all mean that UK is more powerful. Similarly, a US dollar approximately exchange for 122 Japanese yen currently. By no imagination, it means that the US is multiple times more powerful than Japan.

What if the US devalued the dollar the way the Chinese devalued their currency?

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What if the US devalued the dollar the way the Chinese devalued their currency?
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China’s total trade (exports+imports) as percentage of GDP is about twice that of the US. In this sense, the US is a relatively closed economy and therefore its monetary policy is (and ought to be) more domestically oriented. Second, Chinese trade with the US is only about 15% of the US trade with the rest of the world. Any exchange rate intervention by the US will thus also affect the rest of the world, so yes, any such step will force the other countries to follow suit and lead to a zero sum currency war as you rightly talked about.

Why can't central banks just print money and turn it over to their national governments to generate inflation?

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Why can't central banks just print money and turn it over to their national governments to generate inflation?
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In the US at least, financing Govt expenditure/budget deficit by using more Federal Reserve notes is not an “usual” option (the way it is in India for example) is primarily because the central bank’s independence from the Treasury and the Govt’s commitment to let the central bank stay independent.

Would it be bad if rich countries print money and instead of doing the QE inserted the money on a bottom up approach?

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Would it be bad if rich countries print money and instead of doing the QE inserted the money on a bottom up approach?
Answer: 

There are several issues here. First, the US central bank, the Fed, is an independent monetary authority and does what it thinks best to keep inflation and unemployment low. They cannot be "asked" to print more money or change interest rates. Second, while there is considerable merit in your suggestion to print money and give them to poorer segments of society, there are easier ways to achieve the same, via tax cuts or the earned income tax credit or higher minimum wages.