Question

Prove that the following is irrational.
$3+2\sqrt{5}$

Answer 1

Let us assume that $3+2\sqrt{5}$ is rational number

Hence $3+2\sqrt{5}$ can be written in the form of $\frac{a}{b}$ where $a,b(b\ne 0)$ are co-prime

$⟹3+2\sqrt{5}=\frac{a}{b}$

$⟹2\sqrt{5}=\frac{a}{b}-3$

$⟹\frac{a-3b}{2b}=\sqrt{5}$

But here $\sqrt{5}$ is irrational and $\frac{a-3b}{2b}$ is rational

as $Rational\ne Irrational$

This is a contradiction

so $3+2\sqrt{5}$ is a irrational number