Simplify:- '(137*137 + 137*133 + 133*133) / (137*137*137 - 133*133*133)'

Options

A. a. 3/2
B. b. 1/2
C. c. 1/3
D. d. 1/4

Answer

To simplify the expression \((137 \times 137 + 137 \times 133 + 133 \times 133) / (137 \times 137 \times 137 - 133 \times 133 \times 133)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{137^2 + 137 \times 133 + 133^2}{137^3 - 133^3} \]

### Step 2: Identify the numerator and denominator
The numerator can be recognized as a quadratic expression in terms of \(137\) and \(133\):
\[ A^2 + AB + B^2 \quad \text{where } A = 137 \text{ and } B = 133 \]
The denominator can be factored using the difference of cubes:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]

### Step 3: Factor the denominator
Using the difference of cubes formula:
\[ 137^3 - 133^3 = (137 - 133)(137^2 + 137 \times 133 + 133^2) \]
This simplifies to:
\[ 4(137^2 + 137 \times 133 + 133^2) \]

### Step 4: Substitute back into the expression
Now we can substitute this back into our original expression:
\[ \frac{137^2 + 137 \times 133 + 133^2}{4(137^2 + 137 \times 133 + 133^2)} \]

### Step 5: Cancel the common terms
The \(137^2 + 137 \times 133 + 133^2\) in the numerator and denominator cancels out:
\[ \frac{1}{4} \]

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

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