Question

Simplify the following:

`sqrt(7 + 4sqrt(3))`

Answer 1

Given: `sqrt(7 + 4sqrt(3))`

Step wise calculation:

1. Assume the expression can be written as `sqrt(7 + 4sqrt(3)) = sqrt(a) + sqrt(b)` where (a) and (b) are positive real numbers.

2. Squaring both sides: `7 + 4sqrt(3)` 

= `(sqrt(a) + sqrt(b))^2` 

= `a + b + 2sqrt(ab)`

3. Equate rational and irrational parts: a + b = 7

`2sqrt(ab) = 4sqrt(3)` 

⇒ `sqrt(ab) = 2sqrt(3)`

⇒ ab = 4 × 3

⇒ ab = 12

4. Solve the system: a + b = 7, ab = 12.

5. The numbers (a) and (b) are roots of x² – 7x + 12 = 0.

6. Solving quadratic:

`x = (7 +- sqrt(49 - 48))/2`

= `(7 +- 1)/2`

So, x = 4 or x = 3

Thus, a = 4, b = 3 or vice versa.

`sqrt(7 + 4sqrt(3))`

= `sqrt(4) + sqrt(3)`

= `2 + sqrt(3)`

Hence, `sqrt(7 + 4sqrt(3)) = 2 + sqrt(3)`.