If `a = (sqrt3 + sqrt2) ^(-3) and b = ( sqrt3 - sqrt2)^(-3),` then the value of `(a + 1) ^(-1) + (b + 1) ^(-1)` is :

Options

A. `48 sqrt2`
B. `50 sqrt3`
C. `1`
D. `5`

Answer

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

### Step-by-step Solution:

1. **Calculate \(a\) and \(b\)**:
\[ a = (\sqrt{3} + \sqrt{2})^{-3} = \frac{1}{(\sqrt{3} + \sqrt{2})^3} \]
\[ b = (\sqrt{3} - \sqrt{2})^{-3} = \frac{1}{(\sqrt{3} - \sqrt{2})^3} \]

2. **Find \( (a + 1)^{-1} \) and \( (b + 1)^{-1} \)**:
\[ (a + 1)^{-1} = \frac{1}{a + 1} = \frac{1}{\frac{1}{(\sqrt{3} + \sqrt{2})^3} + 1} \]
\[ (b + 1)^{-1} = \frac{1}{b + 1} = \frac{1}{\frac{1}{(\sqrt{3} - \sqrt{2})^3} + 1} \]

3. **Combine the two fractions**:
\[ (a + 1)^{-1} + (b + 1)^{-1} = \frac{1}{\frac{1}{(\sqrt{3} + \sqrt{2})^3} + 1} + \frac{1}{\frac{1}{(\sqrt{3} - \sqrt{2})^3} + 1} \]

4. **Simplify each term**:
\[ (a + 1)^{-1} = \frac{(\sqrt{3} + \sqrt{2})^3}{1 + (\sqrt{3} + \sqrt{2})^3} \]
\[ (b + 1)^{-1} = \frac{(\sqrt{3} - \sqrt{2})^3}{1 + (\sqrt{3} - \sqrt{2})^3} \]

5. **Find a common denominator**:
\[ \text{Common denominator} = (1 + (\sqrt{3} + \sqrt{2})^3)(1 + (\sqrt{3} - \sqrt{2})^3) \]

6. **Combine the numerators**:
\[ \text{Numerator} = (\sqrt{3} + \sqrt{2})^3(1 + (\sqrt{3} - \sqrt{2})^3) + (\sqrt{3} - \sqrt{2})^3(1 + (\sqrt{3} + \sqrt{2})^3) \]

7. **Simplify the expression**:
After some algebraic manipulations, we can find that the numerator simplifies down to a constant value.

8. **Final Calculation**:
After simplifying, we find that:
\[ (a + 1)^{-1} + (b + 1)^{-1} = 1 \]

### Final Answer:
The value of \((a + 1)^{-1} + (b + 1)^{-1}\) is \(1\).