Prove that √p+√q is irrational,where p and q are prime.

1.	Prove that √p+√q is irrational,where p and q are prime.
Answer» Consider\xa0{tex}\\sqrt { p } + \\sqrt { q }{/tex}\xa0is rational and can be represented as\xa0{tex}\\sqrt { p } + \\sqrt { q }{/tex}\xa0= a{tex}\\Rightarrow ( \\sqrt { p } ) = a - \\sqrt { q }{/tex}{tex}\\Rightarrow ( \\sqrt { p } ) ^ { 2 } = ( a - \\sqrt { q } ) ^ { 2 }{/tex}\xa0(squaring both sides)⇒ p = a2 + {tex}\\left(\\sqrt q\\right)^2{/tex} - 2 a {tex}\\sqrt { q }{/tex}⇒ p = a2 + q - 2 a {tex}\\sqrt { q }{/tex}⇒ 2a {tex} \\sqrt { q }{/tex} = a2 + q - p{tex}\\Rightarrow \\sqrt { q } = \\frac { a ^ { 2 } + q - p } { 2 a }{/tex}As q is prime so\xa0{tex}\\sqrt { q }{/tex}\xa0is not rational but\xa0{tex}\\frac { a ^ { 2 } + q - p } { 2 a }{/tex}\xa0is rational because a, p, q are non-zero integers which contradicts our consideration.Hence,\xa0{tex}\\sqrt { p } + \\sqrt { q }{/tex}\xa0is irrational where p and q are primes.

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