Question

Prove the following are irrational.
$6+\sqrt{2}$

Answer 1

Let contrary that $6+\sqrt{2}$ be a rational number

Then we can write

$6+\sqrt{2}=\frac{a}{b}$, where $a$ & $b$

are co-primw integers & $b\ne 0$

$\Rightarrow \sqrt{2}=\frac{a}{b}-6$

$\Rightarrow \sqrt{2}=\frac{a-6}{b}$

Since $a$ & $b$ are integers

So $\frac{a-6}{b}$ is rational

$\Rightarrow \sqrt{2}$ is rational

this contradict the fact that $\sqrt{2}$ is irrational

Therefore our assumption is wrong

Hence $6+\sqrt{2}$ is an irrational number (Proved)