Question

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

Answer 1

Let us suppose `sqrt(3) + sqrt(5)` is rational.

Let `sqrt(3) + sqrt(5)` = a, where a is rational.

Therefore, `sqrt(3) = a - sqrt(5)` 

On squaring both sides, we get

`(sqrt(3))^2 = (a - sqrt(5))^2`

`\implies` 3 = `a^2 + 5 - 2a sqrt(5)`   ......[∵ (a – b)² = a² + b² – 2ab]

`\implies` `2a sqrt(5) = a^2 + 2`

Therefore, `sqrt(5) = (a^2 + 2)/(2a)`, which is a contradiction as the right-hand side is rational number while `sqrt(5)` is irrational.

Hence, `sqrt(3) + sqrt(5)` is irrational.