Question

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

Answer 1

Determine the proving of $3+2\sqrt{5}$that is irrational.

let's take that $3+2\sqrt{5}$as a rational number so, we can write this answer as $3+2\sqrt{5}=\frac{a}{b}$
​Here $aandb$ use two coprime numbers and $b\ne 0$.

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

Here $aandb$ are integers so $\frac{a-3b}{2b}$it is a rational number so $\sqrt{5}$should be a rational number but $\sqrt{5}$ is an irrational number so it contradicts.
Hence, the $3+2\sqrt{5}$ is irrational.