If `x = 7 + 2 sqrt10,` then what is the value of `sqrtx - (1)/(sqrtx)` ?

Options

A. `2 sqrt2`
B. `(2)/(3) (2 sqrt5 + sqrt2)`
C. `- 2 sqrt2`
D. `(2)/(3) ( 2sqrt2 + sqrt5)`

Answer

To solve the problem, we need to find the value of \( \sqrt{x} - \frac{1}{\sqrt{x}} \) given that \( x = 7 + 2\sqrt{10} \).

### Step-by-Step Solution:

1. **Calculate \( \sqrt{x} \)**:
We start with \( x = 7 + 2\sqrt{10} \). To find \( \sqrt{x} \), we can express \( \sqrt{x} \) in a more manageable form. We will assume \( \sqrt{x} = \sqrt{a} + \sqrt{b} \) for some \( a \) and \( b \).

Let's set \( \sqrt{x} = \sqrt{5} + \sqrt{2} \).

To verify this, we need to square it:
\[ (\sqrt{5} + \sqrt{2})^2 = 5 + 2 + 2\sqrt{5}\sqrt{2} = 7 + 2\sqrt{10} \]
This matches our original \( x \), so:
\[ \sqrt{x} = \sqrt{5} + \sqrt{2} \]

2. **Calculate \( \frac{1}{\sqrt{x}} \)**:
Now we need to find \( \frac{1}{\sqrt{x}} \):
\[ \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{5} + \sqrt{2}} \]
To simplify this, we multiply the numerator and the denominator by the conjugate \( \sqrt{5} - \sqrt{2} \):
\[ \frac{1}{\sqrt{5} + \sqrt{2}} \cdot \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{\sqrt{5} - \sqrt{2}}{(\sqrt{5})^2 - (\sqrt{2})^2} = \frac{\sqrt{5} - \sqrt{2}}{5 - 2} = \frac{\sqrt{5} - \sqrt{2}}{3} \]

3. **Combine the results**:
Now we can substitute back into our expression:
\[ \sqrt{x} - \frac{1}{\sqrt{x}} = (\sqrt{5} + \sqrt{2}) - \frac{\sqrt{5} - \sqrt{2}}{3} \]
To combine these, we need a common denominator. The common denominator is 3:
\[ = \frac{3(\sqrt{5} + \sqrt{2}) - (\sqrt{5} - \sqrt{2})}{3} \]
Simplifying the numerator:
\[ = \frac{3\sqrt{5} + 3\sqrt{2} - \sqrt{5} + \sqrt{2}}{3} = \frac{(3\sqrt{5} - \sqrt{5}) + (3\sqrt{2} + \sqrt{2})}{3} = \frac{2\sqrt{5} + 4\sqrt{2}}{3} \]

4. **Final Result**:
Thus, the value of \( \sqrt{x} - \frac{1}{\sqrt{x}} \) is:
\[ \frac{2\sqrt{5} + 4\sqrt{2}}{3} \]