Question

Prove that 2 + 3 root 3 is an irrational number when it is given that root 3 is an irrational number .

Answer 1

Suppose 2+ √3 is rational number, i.e,

$2 + \sqrt{3} = \frac{p}{q} \\ \\ 2q \: + \sqrt{3} q = p \\ \\ \sqrt{3} q = p - 2q \\ \\ \sqrt{3} = \frac{p - 2q}{q}$

We can see that RHS is rational number but LHS is irrational. Which is a contradiction.

Therefore our supposition is wrong. Hence, 2+√3 is irrational number.

Hope it helps.