If P is a prime and x is a real number such that x^2 = P, then x is irrational. Solution Let, x be rational So, x=r/s ,r,s integers, gcd(r,s)=1 SO, (r/s)^2=P r^2=Ps^2 Henec, P|r^2 P is prime so , P|r Hence, r=kP for some integer k k^2P^2=Ps^2 Pk^2=s^2 So, P|s s=tP for some integer t But, gcd(r,s)=1 HEnce a contradiction HEnce, x is irrational.