3+2 under root 5 is irrational prove that

1.	3+2 under root 5 is irrational prove that
Answer» We will prove this by contradiction.Let us suppose that (3+2 {tex}\\sqrt { 5 }{/tex}) is rational.It means that we have co-prime integers a and b such that{tex}\\frac { a } { b } = 3 + 2 \\sqrt { 5 } \\quad \\frac { a } { b } - 3 = 2 \\sqrt { 5 }{/tex}{tex}\\Rightarrow \\frac{{a - 3b}}{b} = 2{\\sqrt 5 \\,{ \\Rightarrow }}\\frac{{a - 3b}}{{2b}} = \\sqrt 5 {/tex} ....(1)a and b are integers.It means L.H.S of (1) is rational but we know that {tex}\\sqrt { 5 }{/tex} is irrational. It is not possible. Therefore, our supposition is wrong. (3+2 {tex}\\sqrt { 5 }{/tex}) cannot be rational.Hence, (3+2 {tex}\\sqrt { 5 }{/tex}) is irrational.

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