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QUESTION
Is the square root of 14 rational?
No, ##sqrt(14) = sqrt(2*7)## is irrational for the same reasons that ##sqrt(15)## is irrational.
http://socratic.org/questions/how-do-you-prove-that-square-root-15-is-irrational
See my proof for ##sqrt(15)##.
If the radicand (the expression under the square root sign) only has perfect square factors then the square root is rational. Otherwise it will be irrational.
