Question

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

Answer 1

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

$\to$ let take that $3+2\sqrt{5}$is rational number

$\to$ so, we can write this answer as

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

Here $a$ & $b$ use two coprime number and $b\ne 0.$

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

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

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

Here $a$ and $b$ are integer so $\frac{a-3b}{2b}$ is a rational number so $\sqrt{5}$ should be rational number but

$\sqrt{5}$ is a irrational number so it is contradict

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