Question

Simplify the following:

`sqrt(7 - 2sqrt(10))`

Answer 1

Given: `sqrt(7 - 2sqrt(10))`

Step-wise calculation:

1. Assume the expression can be written in the form `sqrt(7 - 2sqrt(10)) = sqrt(a) - sqrt(b)` where (a) and (b) are positive real numbers.

2. Square both sides:

`7 - 2sqrt(10)`

= `(sqrt(a) - sqrt(b))^2`

= `a + b - 2sqrt(ab)` 

3. Equate the rational and irrational parts: 

a + b = 7 and `2sqrt(ab) = 2sqrt(10)` which implies:

`sqrt(ab) = sqrt(10)`

⇒ ab = 10

4. Solve the system: a + b = 7 

ab = 10

5. Consider (a) and (b) as roots of quadratic equation: x² – 7x + 10 = 0.

6. Solve quadratic:

`x = (7 +- sqrt(49 - 40))/2`

`x = (7 +- 3)/2`

So, x = 5 or x = 2.

7. Thus, a = 5 and b = 2 or vice versa.

8. Substitute back: `sqrt(7 - 2sqrt(10)) = sqrt(5) - sqrt(2)`.