Question

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

Answer 1

Given: `sqrt(3)` and `sqrt(5)` are irrational numbers.

To Prove: `sqrt(3) + sqrt(5)` is an irrational number.

Proof:

1. Assume the contrary: Suppose `sqrt(3) + sqrt(5) = a`, where (a) is a rational number.

2. From this assumption, express `sqrt(3)` as `sqrt(3) = a - sqrt(5)`

3. Square both sides:

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

`3 = a^2 + 5 - 2asqrt(5)`

4. Simplify the equation:

`3 = a^2 + 5 - 2asqrt(5)`

⇒ `3 - a^2 - 5 = -2asqrt(5)`

⇒ `-(a^2 + 2) = -2asqrt(5)`

⇒ `a^2 + 2 = 2asqrt(5)`

5. Isolate `sqrt(5)`:

`sqrt(5) = (a^2 + 2)/(2a)`

6. Since (a) is rational, `(a^2 + 2)/(2a)` is a rational number.

7. But `sqrt(5)` is irrational. This contradicts the fact that a rational number equals an irrational.

8. Hence, the assumption that `sqrt(3) + sqrt(5)` is rational is wrong.

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