If p,q are prime positive integers prove √p+√q is irrational

1.	If p,q are prime positive integers prove √p+√q is irrational
Answer» Suppose that {tex} \\sqrt { p } + \\sqrt { q }{/tex} is a rational number equal to {tex} \\frac { a } { b }{/tex}, where a and b are integers having no common factor.Now,\xa0{tex} \\sqrt { p } + \\sqrt { q } = \\frac { a } { b }{/tex}{tex} \\Rightarrow \\sqrt { p } = \\frac { a } { b } - \\sqrt { q }{/tex} (squaring both side){tex} \\Rightarrow \\quad ( \\sqrt { p } ) ^ { 2 } = \\left( \\frac { a } { b } - \\sqrt { q } \\right) ^ { 2 }{/tex}{tex} \\Rightarrow \\quad p = \\frac { a ^ { 2 } } { b ^ { 2 } } - 2 \\left( \\frac { a } { b } \\right) \\sqrt { q } + q{/tex}{tex} \\Rightarrow \\quad 2 \\left( \\frac { a } { b } \\right) \\sqrt { q } = \\frac { a ^ { 2 } } { b ^ { 2 } } + q - p{/tex}{tex} \\Rightarrow \\quad 2 \\frac { a } { b } \\sqrt { q } = \\frac { a ^ { 2 } + b ^ { 2 } ( q - p ) } { b ^ { 2 } }{/tex}{tex} \\Rightarrow \\quad \\sqrt { q } = \\frac { a ^ { 2 } + b ^ { 2 } ( q - p ) } { 2 a b }{/tex}{tex} \\Rightarrow \\sqrt { q }{/tex}\xa0is a rational number. (because sum of two rational numbers is always rational)This is a contradiction as\xa0{tex} \\sqrt { q }{/tex}\xa0is an irrational number.Hence,\xa0{tex} \\sqrt { p } + \\sqrt { q }{/tex}\xa0is an irrational number.

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