`(137 xx 137 + 137 xx 133 + 133 xx 133)/(137 xx 137 xx 137 - 133 xx 133 xx 133)` is equal to

Options

A. 4
B. 270
C. `(1)/(4)`
D. `(1)/(270)`

Answer

To solve the expression \((137 \times 137 + 137 \times 133 + 133 \times 133)/(137 \times 137 \times 137 - 133 \times 133 \times 133)\), we can simplify it step by step.

### Step 1: Identify Variables
Let \( A = 137 \) and \( B = 133 \).

### Step 2: Rewrite the Expression
Now, we can rewrite the expression using \( A \) and \( B \):
\[ \frac{A^2 + A \cdot B + B^2}{A^3 - B^3} \]

### Step 3: Simplify the Numerator
The numerator \( A^2 + A \cdot B + B^2 \) can be factored using the identity:
\[ A^2 + A \cdot B + B^2 = (A + B)^2 - AB \]
Calculating \( A + B \) and \( AB \):
\[ A + B = 137 + 133 = 270 \]
\[ AB = 137 \times 133 \]

Now we can compute \( A^2 + A \cdot B + B^2 \):
\[ A^2 + A \cdot B + B^2 = 270^2 - (137 \times 133) \]

### Step 4: Simplify the Denominator
The denominator \( A^3 - B^3 \) can be factored using the difference of cubes:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
Calculating \( A - B \):
\[ A - B = 137 - 133 = 4 \]

### Step 5: Substitute Back
Now we can substitute back into the expression:
\[ \frac{(A^2 + A \cdot B + B^2)}{(A - B)(A^2 + AB + B^2)} = \frac{(A^2 + A \cdot B + B^2)}{4(A^2 + AB + B^2)} \]

### Step 6: Cancel Common Terms
Since \( A^2 + A \cdot B + B^2 \) appears in both the numerator and denominator, we can cancel it out:
\[ \frac{1}{4} \]

### Final Answer
Thus, the expression simplifies to:
\[ \frac{1}{4} \]