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प्रश्न
Prove that \[\sqrt{5} + \sqrt{3}\] is irrational.
उत्तर
Let us assume that \[\sqrt{5} + \sqrt{3}\] is rational .Then, there exist positive co primes a and b such that
`sqrt5+sqrt3=a/b`
`sqrt5=a/b-sqrt3`
`(sqrt5)^2-(2asqrt3)/b+3`
`5= (a/b)^2-(2asqrt3)/b`
`⇒ 5-3=(a/b)^2-(2asqrt3)/b`
`⇒ 2= (a/b)^20-(2asqrt3)/b`
`⇒(a/b)^2-2=(2asqrt3)/b`
`⇒(a^2-2b^2)/b^2=(2asqrt3)/b`
`⇒ ((a^2-2b^2)/b^2)(b/(2a))=sqrt3`
`⇒ sqrt3= ((a^2-2b^2)/(2ab))`
Here we see that `sqrt3` is a rational number which is a contradiction as we know that `sqrt3`is an irrational number.
Hence `sqrt5+sqrt3` is irrational
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