If ` x gt 1 and x ^(2) + (1)/(x^(2)) = 83` then ` x^(3) - (1)/( x^(3) ` is

Options

A. 764
B. 750
C. 756
D. 760

Answer

To solve the problem, we need to find the value of \( x^3 - \frac{1}{x^3} \) given that \( x > 1 \) and \( x^2 + \frac{1}{x^2} = 83 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x^2 + \frac{1}{x^2} = 83 \]

2. **Subtract 2 from both sides:**
\[ x^2 + \frac{1}{x^2} - 2 = 83 - 2 \]
\[ x^2 + \frac{1}{x^2} - 2 = 81 \]

3. **Recognize the identity:**
The left side can be rewritten using the identity \( (x - \frac{1}{x})^2 = x^2 - 2 + \frac{1}{x^2} \):
\[ (x - \frac{1}{x})^2 = 81 \]

4. **Take the square root of both sides:**
\[ x - \frac{1}{x} = 9 \quad \text{(since \( x > 1 \), we take the positive root)} \]

5. **Let \( k = x - \frac{1}{x} \):**
\[ k = 9 \]

6. **Use the identity for \( x^3 - \frac{1}{x^3} \):**
The formula is:
\[ x^3 - \frac{1}{x^3} = (x - \frac{1}{x})^3 + 3(x - \frac{1}{x}) \]
Substituting \( k \):
\[ x^3 - \frac{1}{x^3} = k^3 + 3k \]

7. **Calculate \( k^3 \) and \( 3k \):**
\[ k^3 = 9^3 = 729 \]
\[ 3k = 3 \times 9 = 27 \]

8. **Combine the results:**
\[ x^3 - \frac{1}{x^3} = 729 + 27 = 756 \]

### Final Answer:
\[ x^3 - \frac{1}{x^3} = 756 \]