Question

Prove that `(sqrt(2) + sqrt(5))` is irrational.

Answer 1

Given: Let `x = sqrt(2) + sqrt(5)`

To Prove: x is irrational

Proof [Step-wise]:

1. Assume rationality

Assume for the sake of contradiction that `sqrt(2) + sqrt(5)` is a rational number.

By definition, we can set it equal to a rational variable x.

`x = sqrt(2) + sqrt(5)`

2. Isolate a radical

Isolate one of the square root terms by subtracting `sqrt(2)` from both sides of the equation.

`x - sqrt(2) = sqrt(5)`

3. Square both sides

Square both sides of the equation to eliminate the radical on the right side.

`(x - sqrt(2))^2 = (sqrt(5))^2`

Expand the left side using the algebraic identity (a – b)² = a² – 2ab + b².

`x^2 - 2xsqrt(2) + 2 = 5`

4. Isolate the remaining radiacl

Rearrange the equation to isolate the remaining term containing `sqrt(2)`. First, subtract 2 from both sides.

`x^2 - 2xsqrt(2) = 3`

Next, move x² to the right side.

`-2xsqrt(2) = 3 - x^2`

Multiply the entire equation by –1 to make the radical term positive.

`2xsqrt(2) = x^2 - 3`

Finally, divide both sides by 2x. Note that x ≠ 0 because `sqrt(2) + sqrt(5)` is clearly greater than 0.

`sqrt(2) = (x^2 - 3)/(2x)`

5. Identify the contradiction

Analyze the nature of both sides of the final equation:

Right side `((x^2 - 3)/(2x))`: Since x is assumed to be rational number, its square (x²), subtraction by 3 (x² – 3) and division by 2x must all result in a rational number.

Left side`(sqrt(2))`: It is a well-established mathematical fact that `sqrt(2)` is an irrational number.

An irrational number cannot equal a rational number. This contradiction means our initial assumption must be false.

Since assuming `sqrt(2) + sqrt(5)` is rational leads to a direct mathematical contradiction, the original number `sqrt(2) + sqrt(5)` must be irrational.