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

Answer

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

### Step 1: Find \( \frac{1}{x} \)
We start by calculating \( \frac{1}{x} \) to help us later in finding \( x^3 - \frac{1}{x^3} \).

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

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

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

Calculating the denominator:

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

So we have:

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

### Step 2: Calculate \( x - \frac{1}{x} \)
Now we can find \( x - \frac{1}{x} \):

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

### Step 3: Use the identity for \( x^3 - \frac{1}{x^3} \)
We will use the identity:

\[ x^3 - \frac{1}{x^3} = \left( x - \frac{1}{x} \right)^3 + 3\left( x - \frac{1}{x} \right) \]

Substituting \( x - \frac{1}{x} = 4\sqrt{2} \):

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

### Step 4: Calculate \( (4\sqrt{2})^3 \)
Calculating \( (4\sqrt{2})^3 \):

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

### Step 5: Calculate \( 3(4\sqrt{2}) \)
Calculating \( 3(4\sqrt{2}) \):

\[ 3(4\sqrt{2}) = 12\sqrt{2} \]

### Step 6: Combine the results
Now we combine the results:

\[ x^3 - \frac{1}{x^3} = 128\sqrt{2} + 12\sqrt{2} = 140\sqrt{2} \]

### Final Answer
Thus, the final answer is:

\[ x^3 - \frac{1}{x^3} = 140\sqrt{2} \]

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