What is the value of `(2+sqrt(2))+((1)/(2+sqrt(2))+((1)/(2-sqrt(2)))+(2-sqrt(2))` ?

Options

A. 2
B. 4
C. 8
D. 6

Answer

To solve the expression \((2+\sqrt{2})+\left(\frac{1}{2+\sqrt{2}}+\frac{1}{2-\sqrt{2}}\right)+(2-\sqrt{2})\), we will break it down step by step.

### Step 1: Simplify the Expression
Start with the expression:
\[ (2+\sqrt{2})+\left(\frac{1}{2+\sqrt{2}}+\frac{1}{2-\sqrt{2}}\right)+(2-\sqrt{2}) \]

### Step 2: Combine Like Terms
Combine the terms \( (2+\sqrt{2}) \) and \( (2-\sqrt{2}) \):
\[ (2+\sqrt{2}) + (2-\sqrt{2}) = 2 + 2 + \sqrt{2} - \sqrt{2} = 4 \]
So now we have:
\[ 4 + \left(\frac{1}{2+\sqrt{2}}+\frac{1}{2-\sqrt{2}}\right) \]

### Step 3: Find a Common Denominator for the Fractions
To combine the fractions, we need a common denominator:
\[ \frac{1}{2+\sqrt{2}} + \frac{1}{2-\sqrt{2}} = \frac{(2-\sqrt{2}) + (2+\sqrt{2})}{(2+\sqrt{2})(2-\sqrt{2})} \]

### Step 4: Simplify the Numerator
The numerator simplifies as follows:
\[ (2-\sqrt{2}) + (2+\sqrt{2}) = 2 - \sqrt{2} + 2 + \sqrt{2} = 4 \]

### Step 5: Simplify the Denominator
Now simplify the denominator:
\[ (2+\sqrt{2})(2-\sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2 \]

### Step 6: Combine the Fractions
Now we can combine the fractions:
\[ \frac{4}{2} = 2 \]

### Step 7: Add to the Previous Result
Now add this result to the previous total:
\[ 4 + 2 = 6 \]

### Final Answer
Thus, the value of the entire expression is:
\[ \boxed{6} \]