Suppose . Then . Factor both sides into primes. Then the power of 2 on the RHS is even because it is a square, but the power of 2 on the LHS is odd. This contradicts unique factorization.
Assume that is rational.
Then , where are integers, and is in simplest form (since every rational number can be reduced to simplest form).
Since is even, that means is even. So write it as . Then you have
Remembering that is an integer, so is . Also, since 2 is a factor, is even, and so is even. Write it as .
But we said that was already in simplest form.
So we have a contradiction.
This means that can not possibly be rational.
Therefore is irrational.