Suppose sqrt(3) is rational. Then it has a unique factorization of prime numbers.

sqrt(3)=p1*p2*...*pn

sqrt(3)^2=3^1=p1^2*p2^2*...*pn^2

On left hand side exponent is odd and all exponents are even on right hand side.

By uniqueness of factorization of prime we have a contradiction. Thus sqrt(3) is not rational.