The Mathematical Proof Thread

[A Handmaid]

Also, the proof that an irrational number to an irrational number can equal a rational number is one of my favorites.

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Proof:
Consider A=sqrt(2)^sqrt(2). 
If A is rational, we are done. 
If A is irrational, 
A^sqrt(2)=(sqrt(2)^sqrt(2))^sqrt(2)                      (x^y)^z=x^(y*z)
A^sqrt(2)=sqrt(2)^(sqrt(2)*sqrt(2))
A^sqrt(2)=sqrt(2)^2
A^sqrt(2)=2
And we are done.

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