Prove that (2 + ROOT under 3 )is irrational

1.	Prove that (2 + ROOT under 3 )is irrational
Answer» We will prove this result by contradiction method.Let assume that\xa0{tex}( 2 + \\sqrt { 3 } ){/tex} be rational.Then 2 and {tex}\\sqrt { 3 }{/tex} are rational.{tex}\\Rightarrow 2 + \\sqrt { 3 } - 2{/tex} is rational ..... ({tex}\\because{/tex} difference of two rational numbers\xa0is rational){tex}\\Rightarrow \\sqrt { 3 }{/tex} is rationalThis contradicts the fact that {tex}\\sqrt { 3 }{/tex} is irrational.The Contradiction arises because\xa0we assume that {tex}( 2 + \\sqrt { 3 } ){/tex} is rational.Hence, {tex}2 + \\sqrt { 3 }{/tex} is not a rational but an irrational number.

Discussion