If `x + y = 7` and `xy = 10` , then the value of `((1)/(x^(3)) + (1)/(y^(3)))` is .

Options

A. `0.543`
B. `0.131`
C. `0.133`
D. `0.453`

Answer

To solve the problem, we need to find the value of \( \frac{1}{x^3} + \frac{1}{y^3} \) given that \( x + y = 7 \) and \( xy = 10 \).

### Step-by-step solution:

1. **Use the identity for the sum of cubes**:
\[ \frac{1}{x^3} + \frac{1}{y^3} = \frac{y^3 + x^3}{x^3 y^3} \]
We need to find \( x^3 + y^3 \) and \( x^3 y^3 \).

2. **Find \( x^3 + y^3 \)** using the identity:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
We already know \( x + y = 7 \) and \( xy = 10 \). We need to find \( x^2 + y^2 \).

3. **Calculate \( x^2 + y^2 \)**:
\[ x^2 + y^2 = (x + y)^2 - 2xy = 7^2 - 2 \cdot 10 = 49 - 20 = 29 \]

4. **Substitute back to find \( x^3 + y^3 \)**:
\[ x^3 + y^3 = (x + y)((x^2 + y^2) - xy) = 7(29 - 10) = 7 \cdot 19 = 133 \]

5. **Find \( x^3 y^3 \)**:
\[ x^3 y^3 = (xy)^3 = 10^3 = 1000 \]

6. **Now substitute these values into the expression**:
\[ \frac{1}{x^3} + \frac{1}{y^3} = \frac{x^3 + y^3}{x^3 y^3} = \frac{133}{1000} \]

7. **Final result**:
\[ \frac{133}{1000} = 0.133 \]

### Conclusion:
The value of \( \frac{1}{x^3} + \frac{1}{y^3} \) is \( 0.133 \).