If `sqrt(2)=1.414`, then find the value of `(1)/(2+sqrt(2))`

Explanation

To find the value of \( \frac{1}{2 + \sqrt{2}} \) given that \( \sqrt{2} = 1.414 \), we will rationalize the denominator. Here are the steps:

### Step-by-Step Solution:

1. **Identify the expression to rationalize**:
We need to rationalize \( \frac{1}{2 + \sqrt{2}} \).

2. **Multiply by the conjugate**:
The conjugate of \( 2 + \sqrt{2} \) is \( 2 - \sqrt{2} \). We multiply the numerator and the denominator by this conjugate:
\[ \frac{1}{2 + \sqrt{2}} \cdot \frac{2 - \sqrt{2}}{2 - \sqrt{2}} = \frac{2 - \sqrt{2}}{(2 + \sqrt{2})(2 - \sqrt{2})} \]

3. **Simplify the denominator**:
The denominator can be simplified using the difference of squares formula:
\[ (2 + \sqrt{2})(2 - \sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2 \]
So, we have:
\[ \frac{2 - \sqrt{2}}{2} \]

4. **Separate the terms in the numerator**:
We can separate the fraction:
\[ \frac{2}{2} - \frac{\sqrt{2}}{2} = 1 - \frac{\sqrt{2}}{2} \]

5. **Substitute the value of \( \sqrt{2} \)**:
Now, substitute \( \sqrt{2} = 1.414 \):
\[ 1 - \frac{1.414}{2} \]

6. **Calculate \( \frac{1.414}{2} \)**:
Performing the division:
\[ \frac{1.414}{2} = 0.707 \]

7. **Final calculation**:
Now, subtract this from 1:
\[ 1 - 0.707 = 0.293 \]

### Final Answer:
Thus, the value of \( \frac{1}{2 + \sqrt{2}} \) is \( 0.293 \).