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

Options

A. 2.1014
B. 2.1973
C. 1.9996
D. 1.9876

Answer

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

### Step-by-Step Solution:

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

2. **Find \( \sqrt{x} \)**:
We need to calculate \( \sqrt{x} \):
\[ \sqrt{x} = \sqrt{1 + \sqrt{2}} \]

3. **Find \( \frac{1}{\sqrt{x}} \)**:
To find \( \frac{1}{\sqrt{x}} \), we first rationalize it:
\[ \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{1 + \sqrt{2}}} \]
To rationalize, multiply the numerator and denominator by \( \sqrt{1 - \sqrt{2}} \):
\[ \frac{1}{\sqrt{1 + \sqrt{2}}} \cdot \frac{\sqrt{1 - \sqrt{2}}}{\sqrt{1 - \sqrt{2}}} = \frac{\sqrt{1 - \sqrt{2}}}{\sqrt{(1 + \sqrt{2})(1 - \sqrt{2})}} = \frac{\sqrt{1 - \sqrt{2}}}{\sqrt{1 - 2}} = \frac{\sqrt{1 - \sqrt{2}}}{\sqrt{-1}} = i\sqrt{1 - \sqrt{2}} \]
However, we will use a different approach to find \( \frac{1}{\sqrt{x}} \) directly.

4. **Calculate \( \sqrt{x} + \frac{1}{\sqrt{x}} \)**:
Let \( y = \sqrt{x} + \frac{1}{\sqrt{x}} \). We can square both sides:
\[ y^2 = \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 = x + 2 + \frac{1}{x} \]

5. **Find \( \frac{1}{x} \)**:
Since \( x = 1 + \sqrt{2} \), we find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{1 + \sqrt{2}} \]
To rationalize:
\[ \frac{1}{1 + \sqrt{2}} \cdot \frac{1 - \sqrt{2}}{1 - \sqrt{2}} = \frac{1 - \sqrt{2}}{(1 + \sqrt{2})(1 - \sqrt{2})} = \frac{1 - \sqrt{2}}{1 - 2} = -(1 - \sqrt{2}) = \sqrt{2} - 1 \]

6. **Substitute back into the equation**:
Now we substitute \( x \) and \( \frac{1}{x} \) into the equation for \( y^2 \):
\[ y^2 = (1 + \sqrt{2}) + 2 + (\sqrt{2} - 1) = 2 + \sqrt{2} + 2 + \sqrt{2} - 1 = 2 + 2\sqrt{2} \]

7. **Final calculation**:
Now we have:
\[ y^2 = 2 + 2\sqrt{2} \]
To find \( y \):
\[ y = \sqrt{2 + 2\sqrt{2}} \]

8. **Simplifying \( y \)**:
We can factor \( 2 \) out:
\[ y = \sqrt{2(1 + \sqrt{2})} = \sqrt{2} \cdot \sqrt{1 + \sqrt{2}} = \sqrt{2} \cdot \sqrt{x} \]

### Conclusion:
Thus, the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \) is \( \sqrt{2 + 2\sqrt{2}} \).