my soln is not as perfect as hsbhatt sir's but it can b useful for mcq

by the binomial sereis

(1+x)ⁿ=1+nx+n(n-1)x²/2 + ........

now taking 2^{root 3} (1+1)^{root 3} we get that in the 2nd term of the expansion we get a multiple of root 3 and we will get it in many other terms and they will surely not cancel each other

also we know that rational +irrational=irrational so we it will be irrational.

hope u got it........