Question

Simplify the following:

`sqrt(10 + 2sqrt(21))`

Answer 1

Given: `sqrt(10 + 2sqrt(21))`

Step-wise calculation:

1. Suppose the expression inside the square root can be written as a perfect square of the form `(sqrt(a) + sqrt(b))^2`, where (a) and (b) are positive numbers.

2. Expand `(sqrt(a) + sqrt(b))^2 = a + b + 2sqrt(ab)`.

3. Equate this to the given expression under the root:

a + b = 10

`2sqrt(ab) = 2sqrt(21)`

⇒ `sqrt(ab) = sqrt(21)`

⇒ ab = 21

4. Now solve the system:

a + b = 10

ab = 21

5. Treat as a quadratic in a² – 10a + 21 = 0.

6. Solve:

`a = (10 +- sqrt(100 - 84))/2`

= `(10 +- sqrt(16))/2`

= `(10 +- 4)/2`

Hence, a = 7 or a = 3.

7. Correspondingly, b = 3 or b = 7.

8. Therefore, `sqrt(10 + 2sqrt(21)) = sqrt(7) + sqrt(3)`.