If ` ( x^(2) + (1)/( x^(2)))= (7)/( 4) " for " x gt 0 ` what is the value of ` ( x^(3) + (1)/( x^(3)))`

Options

A. `( ( 3 sqrt(3)))/( 5)`
B. `( ( 3 sqrt(15)))/( 5)`
C. `( ( 3 sqrt(15)))/( 8)`
D. `( ( 3 sqrt(5)))/( 8)`

Answer

To solve the problem, we need to find the value of \( x^3 + \frac{1}{x^3} \) given that \( x^2 + \frac{1}{x^2} = \frac{7}{4} \) for \( x > 0 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x^2 + \frac{1}{x^2} = \frac{7}{4} \]

2. **Add 2 to both sides:**
\[ x^2 + \frac{1}{x^2} + 2 = \frac{7}{4} + 2 \]
\[ x^2 + \frac{1}{x^2} + 2 = \frac{7}{4} + \frac{8}{4} = \frac{15}{4} \]

3. **Recognize the left side as a perfect square:**
\[ \left( x + \frac{1}{x} \right)^2 = \frac{15}{4} \]

4. **Take the square root of both sides:**
\[ x + \frac{1}{x} = \sqrt{\frac{15}{4}} = \frac{\sqrt{15}}{2} \]

5. **Now, we need to find \( x^3 + \frac{1}{x^3} \). Use the identity:**
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3 \left( x + \frac{1}{x} \right) \]

6. **Substituting the value we found:**
\[ x^3 + \frac{1}{x^3} = \left( \frac{\sqrt{15}}{2} \right)^3 - 3 \left( \frac{\sqrt{15}}{2} \right) \]

7. **Calculate \( \left( \frac{\sqrt{15}}{2} \right)^3 \):**
\[ \left( \frac{\sqrt{15}}{2} \right)^3 = \frac{(\sqrt{15})^3}{2^3} = \frac{15\sqrt{15}}{8} \]

8. **Calculate \( 3 \left( \frac{\sqrt{15}}{2} \right) \):**
\[ 3 \left( \frac{\sqrt{15}}{2} \right) = \frac{3\sqrt{15}}{2} \]

9. **Now, substitute back into the equation:**
\[ x^3 + \frac{1}{x^3} = \frac{15\sqrt{15}}{8} - \frac{3\sqrt{15}}{2} \]

10. **Convert \( \frac{3\sqrt{15}}{2} \) to have a common denominator of 8:**
\[ \frac{3\sqrt{15}}{2} = \frac{12\sqrt{15}}{8} \]

11. **Subtract the two fractions:**
\[ x^3 + \frac{1}{x^3} = \frac{15\sqrt{15}}{8} - \frac{12\sqrt{15}}{8} = \frac{3\sqrt{15}}{8} \]

### Final Answer:
\[ x^3 + \frac{1}{x^3} = \frac{3\sqrt{15}}{8} \]