If `x = (sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2)),` then `x^3 + 1/x^3` is equal to

Options

A. 98
B. 1000
C. 5
D. 970

Answer

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

### Step 1: Simplify \( x \)

We start with:
\[ x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \]

To simplify \( x \), we can multiply the numerator and the denominator by the conjugate of the denominator:
\[ x = \frac{(\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \]

### Step 2: Calculate the denominator

The denominator simplifies as follows:
\[ (\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = 3 - 2 = 1 \]

### Step 3: Calculate the numerator

The numerator simplifies as follows:
\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{6} = 5 + 2\sqrt{6} \]

### Step 4: Combine results

Thus, we have:
\[ x = \frac{5 + 2\sqrt{6}}{1} = 5 + 2\sqrt{6} \]

### Step 5: Find \( \frac{1}{x} \)

Now, we find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{5 + 2\sqrt{6}} \]

To rationalize this, multiply the numerator and denominator by the conjugate:
\[ \frac{1}{x} = \frac{5 - 2\sqrt{6}}{(5 + 2\sqrt{6})(5 - 2\sqrt{6})} \]

Calculating the denominator:
\[ (5 + 2\sqrt{6})(5 - 2\sqrt{6}) = 25 - 24 = 1 \]
Thus,
\[ \frac{1}{x} = 5 - 2\sqrt{6} \]

### Step 6: Calculate \( x + \frac{1}{x} \)

Now we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) = 10 \]

### Step 7: Calculate \( x^2 + \frac{1}{x^2} \)

Using the identity:
\[ x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 \]
we find:
\[ x^2 + \frac{1}{x^2} = 10^2 - 2 = 100 - 2 = 98 \]

### Step 8: Calculate \( x^3 + \frac{1}{x^3} \)

Using the identity:
\[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})(x^2 + \frac{1}{x^2}) - (x + \frac{1}{x}) \]
we find:
\[ x^3 + \frac{1}{x^3} = 10 \cdot 98 - 10 = 980 - 10 = 970 \]

### Final Result

Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{970} \]