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

Answer

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

### Step 1: Find \( \frac{1}{x} \)

Given \( x = 3 + 2\sqrt{2} \), we need to find \( \frac{1}{x} \).

\[ \frac{1}{x} = \frac{1}{3 + 2\sqrt{2}} \]

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 3 - 2\sqrt{2} \):

\[ \frac{1}{x} = \frac{1 \cdot (3 - 2\sqrt{2})}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} \]

### Step 2: Simplify the Denominator

Now, we simplify the denominator using the difference of squares formula \( a^2 - b^2 \):

\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]

Thus, we have:

\[ \frac{1}{x} = 3 - 2\sqrt{2} \]

### Step 3: Calculate \( x - \frac{1}{x} \)

Now we can find \( x - \frac{1}{x} \):

\[ x - \frac{1}{x} = (3 + 2\sqrt{2}) - (3 - 2\sqrt{2}) \]

### Step 4: Simplify the Expression

Simplifying the above expression:

\[ x - \frac{1}{x} = 3 + 2\sqrt{2} - 3 + 2\sqrt{2} = 4\sqrt{2} \]

### Step 5: Calculate \( (x - \frac{1}{x})^3 \)

Now we need to find \( (4\sqrt{2})^3 \):

\[ (4\sqrt{2})^3 = 4^3 \cdot (\sqrt{2})^3 = 64 \cdot 2\sqrt{2} = 128\sqrt{2} \]

### Final Answer

Thus, the final answer is:

\[ (x - \frac{1}{x})^3 = 128\sqrt{2} \]

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