Question

Prove that \[\sqrt{5} + \sqrt{3}\] is irrational.

Answer 1

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