The value of `(3 + sqrt(8)) + (1)/(3 - sqrt(8)) - (6 + 4 sqrt(2))` is

Options

A. 8
B. 1
C. `sqrt(2)`
D. 0

Answer

To simplify the expression \((3 + \sqrt{8}) + \frac{1}{(3 - \sqrt{8})} - (6 + 4\sqrt{2})\), we will follow these steps:

### Step 1: Simplify \(\sqrt{8}\)
First, we simplify \(\sqrt{8}\):
\[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]
So, we can rewrite the expression as:
\[ (3 + 2\sqrt{2}) + \frac{1}{(3 - 2\sqrt{2})} - (6 + 4\sqrt{2}) \]

### Step 2: Combine Like Terms
Next, we combine the constant terms and the terms with \(\sqrt{2}\):
\[ (3 - 6) + (2\sqrt{2} - 4\sqrt{2}) + \frac{1}{(3 - 2\sqrt{2})} \]
This simplifies to:
\[ -3 - 2\sqrt{2} + \frac{1}{(3 - 2\sqrt{2})} \]

### Step 3: Rationalize the Denominator
Now, we need to rationalize the denominator of the fraction \(\frac{1}{(3 - 2\sqrt{2})}\). We do this by multiplying the numerator and denominator by the conjugate of the denominator:
\[ \frac{1 \cdot (3 + 2\sqrt{2})}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})} \]
Calculating the denominator:
\[ (3 - 2\sqrt{2})(3 + 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]
So, the fraction simplifies to:
\[ 3 + 2\sqrt{2} \]

### Step 4: Combine All Terms
Now, we substitute back into our expression:
\[ -3 - 2\sqrt{2} + (3 + 2\sqrt{2}) \]
Combining these:
\[ (-3 + 3) + (-2\sqrt{2} + 2\sqrt{2}) = 0 + 0 = 0 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0} \]