Question

Prove that 5+√3 is an irrational number, given that √3 is irrational​

Answer 1

To Prove :

5+√3 is an irrational number

Solution :

$\tt \dagger \: \: \: \: \: Let \: assume \: 5 + \sqrt{3} \: is \: a \\ \tt rational \: number.$

$\tt \longmapsto 5 + \sqrt{3} = \dfrac{p}{q}$

$\tt\small \red{ Where \: p \: and \: q \: co \: prime \: HCF(p , q) = 1.}$

$\tt\longmapsto \sqrt{3} = \dfrac{p}{q} - 5$

$\tt\longmapsto \sqrt{3} = \dfrac{p - 5q}{q}$

Here, We can see √3 is Irrational number ( Given ) and p - 5q / q is Rational number

So,

$\tt Irrational \red{\neq}Rational.$

Our assumption was wrong 5 + √3 is Irrational number.