Question

Prove that $7+\sqrt{2}$ is irrational

Answer 1

Let us assume that $7+\sqrt{2}$ is a rational number
Then. there exist coprime integers $p$, $q$,$q\ne 0$ such that
$7+\sqrt{2}=\frac{p}{q}$
$=>\sqrt{2}=\frac{p}{q}-7$
Here, $\frac{p}{q}-7$ is a rational number, but $\sqrt{2}$ is a irrational number.
But, a irrational cannot be equal to a rational number.This is a contradiction.
Thus, our assumption is wrong.
Therefore $7+\sqrt{2}$ is a irrational number.