Simplify : `(0.41xx0.41xx0.41+0.69xx0.69xx0.69)/(0.41xx0.41-0.41xx0.69+0.69xx0.69)`

Options

A. `0.28`
B. `1.1`
C. `11`
D. `2.8`

Answer

To simplify the expression \((0.41 \times 0.41 \times 0.41 + 0.69 \times 0.69 \times 0.69) / (0.41 \times 0.41 - 0.41 \times 0.69 + 0.69 \times 0.69)\), we can use the formulas for the sum and difference of cubes.

### Step-by-Step Solution:

1. **Identify the terms**:
- Let \( A = 0.41 \) and \( B = 0.69 \).
- The expression can be rewritten as \((A^3 + B^3) / (A^2 - AB + B^2)\).

2. **Use the identity for sum of cubes**:
- The formula for the sum of cubes is:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
- Thus, we can rewrite the numerator:
\[ A^3 + B^3 = (0.41 + 0.69)(0.41^2 - 0.41 \times 0.69 + 0.69^2) \]

3. **Calculate \( A + B \)**:
- \( A + B = 0.41 + 0.69 = 1.10 \).

4. **Calculate \( A^2 \), \( B^2 \), and \( AB \)**:
- \( A^2 = (0.41)^2 = 0.1681 \)
- \( B^2 = (0.69)^2 = 0.4761 \)
- \( AB = 0.41 \times 0.69 = 0.2829 \)

5. **Substitute into the expression**:
- Now substitute these values into \( A^2 - AB + B^2 \):
\[ 0.41^2 - 0.41 \times 0.69 + 0.69^2 = 0.1681 - 0.2829 + 0.4761 \]

6. **Calculate the denominator**:
- \( 0.1681 - 0.2829 + 0.4761 = 0.3613 \).

7. **Now substitute back into the expression**:
- The expression simplifies to:
\[ \frac{(1.10)(0.3613)}{0.3613} \]

8. **Cancel out \( 0.3613 \)**:
- Since \( 0.3613 \) is in both the numerator and denominator, they cancel out:
\[ = 1.10 \]

### Final Answer:
The simplified expression is \( 1.10 \).