Prove that 2+5√3 is an irtational number, given that √3 is an irrational number

Prove that 2+5√3 is an irtational number, given that √3 is an irra

1.	Prove that 2+5√3 is an irtational number, given that √3 is an irrational number
Answer» Let 2+5√3 is an rational no.2+5√3 = a/b5√3 = a/b - 2So, 5√3 is equal to a/b - 2 And we know that a/b - 2 is an rational numberBut it is a fact that √3 is an irrational no.So, therefore, 2+5√3 is an irrational no. If possible, let us suppose that 2 + 5{tex}\\sqrt{3}{/tex} is\xa0a rational numberThen,\xa0we can write\xa0{tex}2+5\\sqrt{3}=\\frac{p}{q} {/tex}( Where p and q are coprime){tex}\\implies 5\\sqrt{3}=\\frac{p}{q}-2{/tex}{tex}\\implies 5\\sqrt{3}=\\frac{p-2q}{q}{/tex}{tex}\\implies \\sqrt{3}=\\frac{p-2q}{5q} {/tex}{tex}\\implies \\sqrt{3}=\\frac{integer}{integer} {/tex}\xa0(Since p and q are integers){tex}\\implies \\sqrt{3} {/tex}\xa0is rational numberwhich is a contradiction to the given fact that {tex}\\sqrt{3}{/tex}\xa0is irrational.{tex}\\therefore{/tex}\xa02 + 5{tex}\\sqrt{3}{/tex}\xa0cannot be rationalHence, 2 + 5{tex}\\sqrt{3}{/tex}\xa0is irrational.