Question

Show that √5+2√3 is irrational

Answer 1

Let us assume that 5+2√3 is rational

5+2√3 = p/q ( where p and q are co prime)

2√3 = p/q-5

2√3 = p-5q/q

√3 = p-5q/2q

now p , 5 , 2 and q are integers 

∴ p-5q/2q is rational

∴ √3 is rational

but we know that √3 is irrational . This is a contradiction which has arisen due to our wrong assumption.

∴ 5+2√3 is irrational