If `x= 3 + 2 sqrt2`, find
`x- (1)/(x)`

Answer

To solve the problem where \( x = 3 + 2\sqrt{2} \) and we need to find \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Find \( \frac{1}{x} \)
We start with the expression for \( x \):
\[ x = 3 + 2\sqrt{2} \]
To find \( \frac{1}{x} \), we write:
\[ \frac{1}{x} = \frac{1}{3 + 2\sqrt{2}} \]
To simplify this, we will rationalize the denominator.

### Step 2: Rationalize the Denominator
We multiply the numerator and the denominator by the conjugate of the denominator, which is \( 3 - 2\sqrt{2} \):
\[ \frac{1}{x} = \frac{1 \cdot (3 - 2\sqrt{2})}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} \]
Calculating the denominator using the difference of squares:
\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]
Thus, we have:
\[ \frac{1}{x} = 3 - 2\sqrt{2} \]

### Step 3: Substitute \( \frac{1}{x} \) into the Expression
Now we substitute \( \frac{1}{x} \) back into the expression \( x - \frac{1}{x} \):
\[ x - \frac{1}{x} = (3 + 2\sqrt{2}) - (3 - 2\sqrt{2}) \]

### Step 4: Simplify the Expression
Now simplify the expression:
\[ x - \frac{1}{x} = 3 + 2\sqrt{2} - 3 + 2\sqrt{2} \]
The \( 3 \) cancels out:
\[ x - \frac{1}{x} = 2\sqrt{2} + 2\sqrt{2} = 4\sqrt{2} \]

### Final Answer
Thus, the final answer is:
\[ x - \frac{1}{x} = 4\sqrt{2} \]