The value of `{1/((sqrt(2) - 2)) + 1/((sqrt(2) + 2)) + (sqrt(2) + 2)}` is :

Options

A. `2`
B. `4`
C. `2sqrt(2)`
D. None of these

Answer

To solve the expression \( \frac{1}{\sqrt{2} - 2} + \frac{1}{\sqrt{2} + 2} + (\sqrt{2} + 2) \), we will follow these steps:

### Step 1: Simplify the first two fractions

To simplify \( \frac{1}{\sqrt{2} - 2} + \frac{1}{\sqrt{2} + 2} \), we will find a common denominator. The common denominator is \( (\sqrt{2} - 2)(\sqrt{2} + 2) \).

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

### Step 2: Combine the numerators

Now, combine the numerators:

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

So, we have:

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

### Step 3: Simplify the denominator

Next, simplify the denominator:

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

### Step 4: Substitute back into the expression

Now, substitute back into the expression:

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

### Step 5: Add the remaining term

Now we need to add the remaining term \( \sqrt{2} + 2 \):

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

### Final Result

Thus, the value of the entire expression is:

\[ \boxed{2} \]