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

Options

A. `(3 sqrt3)/(5)`
B. `(3 sqrt15)/(5)`
C. `- (9 sqrt2)/(7 sqrt7)`
D. `- ( 3 sqrt18)/(7)`

Answer

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

### Step-by-step Solution:

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

2. **Use the identity to find \( x + \frac{1}{x} \):**
We know that:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Let \( y = x + \frac{1}{x} \). Then:
\[ y^2 - 2 = \frac{4}{7} \]
Rearranging gives:
\[ y^2 = \frac{4}{7} + 2 = \frac{4}{7} + \frac{14}{7} = \frac{18}{7} \]

3. **Take the square root to find \( y \):**
\[ y = \sqrt{\frac{18}{7}} = \frac{3\sqrt{2}}{\sqrt{7}} \]

4. **Now use \( y \) to find \( x^3 + \frac{1}{x^3} \):**
We use the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3\left( x + \frac{1}{x} \right) \]
Substituting \( y \):
\[ x^3 + \frac{1}{x^3} = y^3 - 3y \]

5. **Calculate \( y^3 \):**
\[ y^3 = \left( \frac{3\sqrt{2}}{\sqrt{7}} \right)^3 = \frac{27 \cdot 2\sqrt{2}}{7\sqrt{7}} = \frac{54\sqrt{2}}{7\sqrt{7}} \]

6. **Calculate \( 3y \):**
\[ 3y = 3 \cdot \frac{3\sqrt{2}}{\sqrt{7}} = \frac{9\sqrt{2}}{\sqrt{7}} \]

7. **Combine the results:**
\[ x^3 + \frac{1}{x^3} = \frac{54\sqrt{2}}{7\sqrt{7}} - \frac{9\sqrt{2}}{\sqrt{7}} \]
To combine these fractions, we need a common denominator:
\[ = \frac{54\sqrt{2}}{7\sqrt{7}} - \frac{63\sqrt{2}}{7\sqrt{7}} = \frac{54\sqrt{2} - 63\sqrt{2}}{7\sqrt{7}} = \frac{-9\sqrt{2}}{7\sqrt{7}} \]

### Final Answer:
Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \frac{-9\sqrt{2}}{7\sqrt{7}} \]