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

Explanation

To find the value of \( \frac{1}{2 + \sqrt{2}} \), we can follow these steps:

### Step 1: Rationalize the Denominator
To simplify \( \frac{1}{2 + \sqrt{2}} \), we can multiply the numerator and denominator by the conjugate of the denominator, which is \( 2 - \sqrt{2} \).

\[ \frac{1}{2 + \sqrt{2}} \cdot \frac{2 - \sqrt{2}}{2 - \sqrt{2}} = \frac{2 - \sqrt{2}}{(2 + \sqrt{2})(2 - \sqrt{2})} \]

### Step 2: Simplify the Denominator
Now, we simplify the denominator using the difference of squares formula:

\[ (2 + \sqrt{2})(2 - \sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2 \]

### Step 3: Write the Expression
Now we can write the expression as:

\[ \frac{2 - \sqrt{2}}{2} \]

### Step 4: Split the Fraction
We can split the fraction into two parts:

\[ \frac{2}{2} - \frac{\sqrt{2}}{2} = 1 - \frac{\sqrt{2}}{2} \]

### Step 5: Substitute the Value of \( \sqrt{2} \)
Now, we substitute the value of \( \sqrt{2} \) which is given as \( 1.414 \):

\[ 1 - \frac{1.414}{2} \]

### Step 6: Calculate \( \frac{1.414}{2} \)
Calculating \( \frac{1.414}{2} \):

\[ \frac{1.414}{2} = 0.707 \]

### Step 7: Final Calculation
Now, we can find the final value:

\[ 1 - 0.707 = 0.293 \]

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

### Summary of Steps:
1. Rationalize the denominator.
2. Simplify the denominator using the difference of squares.
3. Write the expression in a simpler form.
4. Substitute the value of \( \sqrt{2} \).
5. Calculate the final result.