If `x = sqrt(3) + sqrt(2)` then the value of `x^(3) - (1)/(x^(3))` is

Options

A. `10 sqrt(2)`
B. `14 sqrt(2)`
C. `22 sqrt(2)`
D. `8 sqrt(2)`

Answer

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

### Step 1: Find \( \frac{1}{x} \)
We start by calculating \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{\sqrt{3} + \sqrt{2}} \]
To simplify this expression, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{x} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} \]
Calculating the denominator:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]
Thus, we have:
\[ \frac{1}{x} = \sqrt{3} - \sqrt{2} \]

### Step 2: Find \( x - \frac{1}{x} \)
Now we can find \( x - \frac{1}{x} \):
\[ x - \frac{1}{x} = (\sqrt{3} + \sqrt{2}) - (\sqrt{3} - \sqrt{2}) = \sqrt{3} + \sqrt{2} - \sqrt{3} + \sqrt{2} = 2\sqrt{2} \]

### Step 3: Calculate \( x^3 - \frac{1}{x^3} \)
We can use the identity:
\[ x^3 - \frac{1}{x^3} = \left(x - \frac{1}{x}\right)\left(x^2 + 1 + \frac{1}{x^2}\right) \]
First, we need to find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2 \]
Substituting \( x - \frac{1}{x} = 2\sqrt{2} \):
\[ x^2 + \frac{1}{x^2} = (2\sqrt{2})^2 + 2 = 8 + 2 = 10 \]

Now substituting back into the equation for \( x^3 - \frac{1}{x^3} \):
\[ x^3 - \frac{1}{x^3} = (2\sqrt{2})(10) = 20\sqrt{2} \]

### Final Answer
Thus, the value of \( x^3 - \frac{1}{x^3} \) is:
\[ \boxed{20\sqrt{2}} \]