Prove that 5+2√3 is an irrational number.

 

Given:

$\sqrt{3}$ is irrational.

To Prove:

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

Proof:

Suppose that $5+2\sqrt{3}$ is a rational number. Then, it can be written in $\frac{p}{q}$ form where integers $p$ and $q$ have no common prime factor.

$5+2\sqrt{3}=\frac{p}{q}$

$\text{Where, HCF}(p,q)=1\text{ and }q\ne 0.$

$2\sqrt{3}=\frac{p}{q}-5$

$2\sqrt{3}=\frac{p-5q}{q}$

$\sqrt{3}=\frac{p-5q}{2q}$

It shows that $\sqrt{3}$ can be written in $\frac{p}{q}$ form which cannot be possible because $\sqrt{3}$ is irrational. So, our assumption is wrong. Hence $4+2\sqrt{3}$ is irrational.