Question

Simplify the following:

`sqrt(3 - 2sqrt(2))`

Answer 1

Given: `sqrt(3 - 2sqrt(2))`

Stepwise calculation:

1. Assume the expression inside the square root can be written as a square of a binomial involving square roots: `3 - 2sqrt(2) = (sqrt(a) - sqrt(b))^2 = a + b - 2sqrt(ab)` where (a) and (b) are positive numbers. 

2. Equate terms: a + b = 3

`2sqrt(ab) = 2sqrt(2)`

⇒ `sqrt(ab) = sqrt(2)`

Squaring: ab = 2

3. Solve the system: a + b = 3

ab = 2

4. The pair (a, b) that satisfies both is a = 1, b = 2 because:

1 + 2 = 3 

1 × 2 = 2 

5. So, `3 - 2sqrt(2) = (sqrt(2) - 1)^2`

6. Taking the square root: `sqrt(3 - 2sqrt(2))`

= `sqrt((sqrt(2) - 1)^2`

= `|\sqrt(2) - 1|`

Since `sqrt(2) ≈ 1.414, sqrt(2) - 1 > 0`,

Thus, `sqrt(3 - 2sqrt(2)) = sqrt(2) - 1`.