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What does the Fed do with the extra money it is printing? 
First of all, just for the sake of clarity, the Federal Reserve doesn't actually "print" money - that's the job of the BEP. What they do is they buy US Treasury bonds - i.e., loan money to the US government. The money they do it with are created "from thin air" - just by adding some numbers in certain accounts, thus it is described as "printing money". The US government then spends the money however it wishes to. The idea is that this money is injected into the economy - since the only way the US government can use the money from these loans is to spend them on buying something or give it to some people that would spend them.  As it is a loan, sometime in the future the US government would pay these loans back, and in this moment the Fed would decide - if they want to "contract" the supply of money back, they just "destroy" the money they've got, by erasing the numbers they created before. They could also do it by selling the bonds they hold on the open market and then again "destroy" the money they got as proceeds, thus lowering the amount of money existing in the economy. This way the Fed can control how much money is out there and thus supposedly influence inflation and economic activity.  The Fed could also inject money in the economy by buying any assets after creating the money - for example, right now they own about a trillion dollars worth of various mortgage-based securities. But since buying specific security would probably give unfair advantage to the issuers and owners of this security, usually US treasury bonds if what they buy. The side effect of increased supply of money denominated in dollars would be, as you noted, devaluation of dollars compared to other currencies.