`(sqrt(2) + (1)/(sqrt(2)))^(2)` is equal to :

Options

A. `2 (1)/(2)`
B. `3 (1)/(2)`
C. `4 (1)/(2)`
D. `5 (1)/(2)`

Answer

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

### Step 1: Apply the formula for the square of a binomial
We can use the identity \((a + b)^{2} = a^{2} + 2ab + b^{2}\).

Let \(a = \sqrt{2}\) and \(b = \frac{1}{\sqrt{2}}\).

### Step 2: Calculate \(a^{2}\)
\[ a^{2} = (\sqrt{2})^{2} = 2 \]

### Step 3: Calculate \(b^{2}\)
\[ b^{2} = \left(\frac{1}{\sqrt{2}}\right)^{2} = \frac{1}{2} \]

### Step 4: Calculate \(2ab\)
\[ 2ab = 2 \cdot \sqrt{2} \cdot \frac{1}{\sqrt{2}} = 2 \cdot 1 = 2 \]

### Step 5: Combine the results
Now, substitute \(a^{2}\), \(b^{2}\), and \(2ab\) back into the binomial expansion:
\[ (\sqrt{2} + \frac{1}{\sqrt{2}})^{2} = a^{2} + 2ab + b^{2} = 2 + 2 + \frac{1}{2} \]

### Step 6: Simplify the expression
Combine the terms:
\[ 2 + 2 + \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2} \]

Thus, the final answer is:
\[ \boxed{\frac{9}{2}} \]