Use proof by contradiction…
Suppose ##sqrt(14)## is rational.
Then ##sqrt(14) = p/q## for some positive integers ##p q## with ##q != 0##.
Without loss of generality we can suppose that ##p## and ##q## are the smallest such pair of integers.
##(p/q)^2 = 14##
So:
##p^2 = 14 q^2##
In particular ##p^2## is even.
If ##p^2## is even then ##p## must be even too so ##p = 2k## for some positive integer ##k##.
So:
##14 q^2 = (2k)^2 = 4 k^2##
Dividing both sides by ##2## we get:
##7 q^2 = 2 k^2##
So ##k^2## must be divisible by ##7##. So ##k## must be divisible by ##7## too. So ##k = 7m## for some positive integer ##m##.
##7 q^2 = 2 (7m)^2 = 7*14m^2##
Divide both sides by ##7## to find:
##q^2 = 14 m^2##
So ##14 = q^2/m^2 = (q/m)^2##
So ##sqrt(14) = q/m##
Now ##m < q < p## contradicting our supposition that ##p## and ##q## are the smallest pair of positive integers such that ##sqrt(14) = p/q##. So our supposition is false and therefore our hypothesis that ##sqrt(14)## is rational is also false.