If `x=sqrt3 +sqrt2` then the value of of `x^(3)-(1)/(x^(3))` is

Options

A. `10sqrt2`
B. `14sqrt2`
C. `22sqrt2`
D. `8sqrt2`

Answer

To solve the problem, we need to find the value of \( x^3 - \frac{1}{x^3} \) given that \( x = \sqrt{3} + \sqrt{2} \).

### Step-by-Step Solution:

1. **Find \( \frac{1}{x} \)**:
\[ x = \sqrt{3} + \sqrt{2} \]
To find \( \frac{1}{x} \), we rationalize it:
\[ \frac{1}{x} = \frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2} \]

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

3. **Use the identity for \( x^3 - \frac{1}{x^3} \)**:
The formula for \( x^3 - \frac{1}{x^3} \) in terms of \( x - \frac{1}{x} \) is:
\[ x^3 - \frac{1}{x^3} = \left( x - \frac{1}{x} \right) \left( x^2 + 1 + \frac{1}{x^2} \right) \]
We first need to find \( x^2 + \frac{1}{x^2} \).

4. **Calculate \( x^2 \)**:
\[ x^2 = (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{3}\sqrt{2} = 5 + 2\sqrt{6} \]

5. **Calculate \( \frac{1}{x^2} \)**:
\[ \frac{1}{x^2} = \left( \frac{1}{\sqrt{3} + \sqrt{2}} \right)^2 = \frac{1}{5 + 2\sqrt{6}} \]
To rationalize:
\[ \frac{1}{5 + 2\sqrt{6}} \cdot \frac{5 - 2\sqrt{6}}{5 - 2\sqrt{6}} = \frac{5 - 2\sqrt{6}}{(5)^2 - (2\sqrt{6})^2} = \frac{5 - 2\sqrt{6}}{25 - 24} = 5 - 2\sqrt{6} \]

6. **Calculate \( x^2 + \frac{1}{x^2} \)**:
\[ x^2 + \frac{1}{x^2} = (5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) = 10 \]

7. **Now substitute back into the formula**:
\[ x^3 - \frac{1}{x^3} = (x - \frac{1}{x}) \left( x^2 + \frac{1}{x^2} \right) = (2\sqrt{2})(10) = 20\sqrt{2} \]

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