Prove that `sqrt3+sqrt5` is irrational

Explanation

Let us suppose that `sqrt3+sqrt5` is rational
Let `sqrt3+sqrt5=a`, where a is rational
Therefore, `sqrt3=a-sqrt5`
On squaring both sides, we get
`(sqrt3)^(2)=(a-sqrt5)^(2)`
`Rightarrow 3=a^(2)+5-2asqrt5` `"["therefore (a-b)^(2)=a^(2)+b^(2)-2ab]`
`Rightarrow 2a sqrt5=a^(2)+2`
`Therefore, sqrt5=(a^(2)+2)/(2a)` which is contracdiction.
As the right hand side is rational number while `sqrt5` is irrational. Since 3 and 5 are prime number. Hence, `sqrt3 +sqrt5` is irrational.