Question

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

Answer 1

Assume that $3+2\sqrt{5}$ is rational. Then,

$3+2\sqrt{5}=\frac{a}{b}$

$\sqrt{5}=\frac{a-3b}{2b}$

This gives that $\frac{a-3b}{2b}$ is rational but $\sqrt{5}$ is an irrational number, therefore, the assumption is wrong.

Hence, $3+2\sqrt{5}$ is irrational.