`(343 xx 343 xx 343 + 257 xx 257 xx 257)/(343 xx 343 - 343 xx 257 + 257 xx 257)` = ?

Options

A. 8600
B. 800
C. 600
D. 2600

Answer

To solve the expression \((343 \times 343 \times 343 + 257 \times 257 \times 257)/(343 \times 343 - 343 \times 257 + 257 \times 257)\), we can utilize the formula for the sum and difference of cubes.

### Step-by-Step Solution:

1. **Identify the Cubes**:
We recognize that \(343\) and \(257\) can be represented as \(a\) and \(b\) respectively:
- Let \(a = 343\)
- Let \(b = 257\)

2. **Use the Sum of Cubes Formula**:
The sum of cubes can be expressed as:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Therefore, we can rewrite the numerator:
\[ 343^3 + 257^3 = (343 + 257)(343^2 - 343 \times 257 + 257^2) \]

3. **Calculate \(a + b\)**:
\[ a + b = 343 + 257 = 600 \]

4. **Calculate \(a^2 - ab + b^2\)**:
- First, calculate \(a^2\):
\[ 343^2 = 117649 \]
- Then calculate \(b^2\):
\[ 257^2 = 66049 \]
- Now calculate \(ab\):
\[ 343 \times 257 = 88151 \]
- Substitute these values into the expression:
\[ a^2 - ab + b^2 = 117649 - 88151 + 66049 = 117649 - 88151 + 66049 = 0 \]

5. **Rewrite the Numerator**:
Therefore, the numerator becomes:
\[ (600)(0) = 0 \]

6. **Calculate the Denominator**:
The denominator is:
\[ 343^2 - 343 \times 257 + 257^2 \]
We already calculated \(343^2\), \(257^2\), and \(343 \times 257\) in the previous steps:
\[ 117649 - 88151 + 66049 = 0 \]

7. **Final Expression**:
Now we have:
\[ \frac{0}{0} \]
This is an indeterminate form, but since both the numerator and denominator yield the same expression, we can conclude that the entire expression simplifies to \(0\).

### Final Answer:
\[ \text{The value of the expression is } 0. \]