`(117 xx 117 xx 117 - 98 xx 98 xx 98)/(117 xx 117 + 117 xx 98 + 98 xx 98) = ?`

Options

A. 215
B. 311
C. 19
D. 29

Answer

To solve the expression \((117 \times 117 \times 117 - 98 \times 98 \times 98) / (117 \times 117 + 117 \times 98 + 98 \times 98)\), we can follow these steps:

### Step 1: Rewrite the expression using variables
Let \( A = 117 \) and \( B = 98 \). This allows us to simplify the notation:
\[ \frac{A^3 - B^3}{A^2 + AB + B^2} \]

### Step 2: Recognize the formulas
We can use the identity for the difference of cubes:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
Using this, we can rewrite the expression:
\[ \frac{(A - B)(A^2 + AB + B^2)}{A^2 + AB + B^2} \]

### Step 3: Cancel the common terms
Since \( A^2 + AB + B^2 \) appears in both the numerator and the denominator, we can cancel these terms:
\[ A - B \]

### Step 4: Substitute back the values of A and B
Now, we substitute back the values of \( A \) and \( B \):
\[ 117 - 98 \]

### Step 5: Calculate the final result
Now we perform the subtraction:
\[ 117 - 98 = 19 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{19} \]