Prove that `sqrt(3)+\ sqrt(5)` is an irrational number.

Explanation

Let `sqrt(3)+\ sqrt(5)` be a rational number.
A rational number can be written in the form of p/q where `p,q` are integers.
`sqrt(3)+\ sqrt(5)` = `p/q`
`sqrt 3 ` =` p/q` - `(sqrt 5)`
Squaring on both sides,
`( sqrt 3)² = (p/q-(sqrt 5)²`
`3 = p²/q²+(sqrt5)²-2(p/q)(sqrt 5)`
`sqrt 5` xx `2p/q = p^2/q^2+5-3`
`sqrt 5 = (p²+2q²)/q² × q/2psqrt 5 = (p²+2q²)/2pq`
`p,q` are integers then `(p²+2q²)/2pq` is a rational number.
Then `sqrt 5` is also a rational number.
But this contradicts the fact that `sqrt 5` is an irrational number.
So, our supposition is false. Therefore, `(sqrt 5+sqrt 3)` is an irrational number.