If `x - sqrt3 - sqrt2 = 0 and y - sqrt3 + sqrt2 =0`, then the value of `(x ^(3) - 20 sqrt2) - (y ^(3) + 2 sqrt2)`

Options

A. 1
B. 3
C. 0
D. 2

Answer

To solve the problem, we will follow these steps:

1. **Identify the values of x and y from the equations**:
- From the equation \( x - \sqrt{3} - \sqrt{2} = 0 \), we can rearrange it to find \( x \):
\[ x = \sqrt{3} + \sqrt{2} \]
- From the equation \( y - \sqrt{3} + \sqrt{2} = 0 \), we can rearrange it to find \( y \):
\[ y = \sqrt{3} - \sqrt{2} \]

2. **Calculate \( x^3 \) and \( y^3 \)**:
- We will use the binomial expansion to calculate \( x^3 \) and \( y^3 \).
- For \( x = \sqrt{3} + \sqrt{2} \):
\[ x^3 = (\sqrt{3} + \sqrt{2})^3 = (\sqrt{3})^3 + 3(\sqrt{3})^2(\sqrt{2}) + 3(\sqrt{3})(\sqrt{2})^2 + (\sqrt{2})^3 \]
\[ = 3\sqrt{3} + 3 \cdot 3 \cdot \sqrt{2} + 3 \cdot \sqrt{3} \cdot 2 + 2\sqrt{2} \]
\[ = 3\sqrt{3} + 9\sqrt{2} + 6\sqrt{3} + 2\sqrt{2} \]
\[ = 9\sqrt{3} + 11\sqrt{2} \]

- For \( y = \sqrt{3} - \sqrt{2} \):
\[ y^3 = (\sqrt{3} - \sqrt{2})^3 = (\sqrt{3})^3 - 3(\sqrt{3})^2(\sqrt{2}) + 3(\sqrt{3})(\sqrt{2})^2 - (\sqrt{2})^3 \]
\[ = 3\sqrt{3} - 9\sqrt{2} + 6\sqrt{3} - 2\sqrt{2} \]
\[ = 9\sqrt{3} - 11\sqrt{2} \]

3. **Substitute \( x^3 \) and \( y^3 \) into the expression**:
- We need to evaluate:
\[ (x^3 - 20\sqrt{2}) - (y^3 + 2\sqrt{2}) \]
- Substitute the values of \( x^3 \) and \( y^3 \):
\[ = (9\sqrt{3} + 11\sqrt{2} - 20\sqrt{2}) - (9\sqrt{3} - 11\sqrt{2} + 2\sqrt{2}) \]
\[ = (9\sqrt{3} + 11\sqrt{2} - 20\sqrt{2}) - (9\sqrt{3} - 9\sqrt{2}) \]
\[ = 9\sqrt{3} + 11\sqrt{2} - 20\sqrt{2} - 9\sqrt{3} + 9\sqrt{2} \]
\[ = (9\sqrt{3} - 9\sqrt{3}) + (11\sqrt{2} - 20\sqrt{2} + 9\sqrt{2}) \]
\[ = 0 + (11 - 20 + 9)\sqrt{2} \]
\[ = 0 + 0\sqrt{2} \]
\[ = 0 \]

4. **Final Result**:
- The value of the expression is \( 0 \).