If ` x = 3 + 2 sqrt(2)` , then the value of ` ( sqrt(x) - (1)/( sqrt(x))) ` is

Options

A. 1
B. 2
C. `2 sqrt(2)`
D. `3 sqrt(3)`

Answer

To solve the problem, we need to find the value of \( \sqrt{x} - \frac{1}{\sqrt{x}} \) given that \( x = 3 + 2\sqrt{2} \).

### Step-by-Step Solution:

1. **Identify \( x \)**:
\[ x = 3 + 2\sqrt{2} \]

2. **Express \( x \) in a different form**:
We can rewrite \( x \) as:
\[ x = (1 + \sqrt{2})^2 \]
This is because:
\[ (1 + \sqrt{2})^2 = 1^2 + 2\cdot1\cdot\sqrt{2} + (\sqrt{2})^2 = 1 + 2\sqrt{2} + 2 = 3 + 2\sqrt{2} \]

3. **Find \( \sqrt{x} \)**:
Taking the square root of both sides:
\[ \sqrt{x} = 1 + \sqrt{2} \]

4. **Calculate \( \frac{1}{\sqrt{x}} \)**:
To find \( \frac{1}{\sqrt{x}} \):
\[ \frac{1}{\sqrt{x}} = \frac{1}{1 + \sqrt{2}} \]
To rationalize the denominator, multiply the numerator and denominator by \( 1 - \sqrt{2} \):
\[ \frac{1}{1 + \sqrt{2}} \cdot \frac{1 - \sqrt{2}}{1 - \sqrt{2}} = \frac{1 - \sqrt{2}}{(1 + \sqrt{2})(1 - \sqrt{2})} = \frac{1 - \sqrt{2}}{1 - 2} = -(1 - \sqrt{2}) = \sqrt{2} - 1 \]

5. **Combine \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \)**:
Now we can substitute back into our original expression:
\[ \sqrt{x} - \frac{1}{\sqrt{x}} = (1 + \sqrt{2}) - (\sqrt{2} - 1) \]
Simplifying this gives:
\[ = 1 + \sqrt{2} - \sqrt{2} + 1 = 2 \]

### Final Answer:
The value of \( \sqrt{x} - \frac{1}{\sqrt{x}} \) is \( 2 \).