Question

prove that 7-2√3 is an irrational number

Answer 1

Let assume that 7-2root3 be rational and rational numbers are in the form of p/q form.
$7 - 2 \sqrt{3} = \frac{p}{q} \\ 7 - \frac{p}{q} = 2 \sqrt{3} \\ \frac{7q - p}{q} = 2 \sqrt{3} \\ \frac{7q - p}{2q} = \sqrt{3}$
We know that root 3 is irrational and not p/q form. It contradicts the statement that it is irrational as it is p/q form.
Subtraction and multiplication of irrational number form irrational number so
7-2root3 is irrational.