Question

If `x = 3 + 2sqrt(2)`, find the value of `sqrt(x) + 1/sqrt(x)`.

Answer 1

Given: `x = 3 + 2sqrt(2)`

We need to find the value of `sqrt(x) + 1/sqrt(x)`.

Step-wise calculation:

1. Let `y = sqrt(x) + 1/sqrt(x)`.

2. Square both sides:

`y^2 = (sqrt(x) + 1/sqrt(x))^2`

`y^2 = x + 2 + 1/x`

3. We already have:

`x = 3 + 2sqrt(2)` and we find `1/x = 1/(3 + 2sqrt(2))`

4. Rationalize the denominator of `1/x`: 

`1/(3 + 2sqrt(2)) = (3 - 2sqrt(2))/((3 + 2sqrt(2))(3 - 2sqrt(2))`

`1/(3 + 2sqrt(2)) = (3 - 2sqrt(2))/(9 - 8)`

`1/(3 + 2sqrt(2)) = 3 - 2sqrt(2)`

5. Now, sum `x + 1/x`:

`(3 + 2sqrt(2)) + (3 - 2sqrt(2)) = 3 + 3 + 2sqrt(2) - 2sqrt(2)`

`(3 + 2sqrt(2)) + (3 - 2sqrt(2)) = 6`

6. Substitute the values back into y²:

`y^2 = x + 2 + 1/x`

y² = 6 + 2

y² = 8

7. Finally,

`y = sqrt(8)`

`y = 2sqrt(2)`