I believe that this is a standard proof but I have a problem with it.

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The end statement appears to me to be

root2 = a/b where a and b are both even therefore a and b are not coprime which contradicts the initial condition that a and b must be coprime therefore root2 is irrational.

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I ran through a different proof to attempt to prove that root4 is irrational

(this is not the full proof, I have truncated it a little)

suppose root4 were irrational

then root4 = a/b

4b^2=a^2

therefore a is even

Let a=2k

4b^2=4k^2

b^2=k^2

k=+-b

therefore

a=2k and b=+/-k

Therefore a and b are not coprime