`(137xx137+133xx133+18221)/(137xx137xx137-133xx133xx133)` is equal to

Options

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

Answer

To solve the expression \((137 \times 137 + 133 \times 133 + 18221) / (137 \times 137 \times 137 - 133 \times 133 \times 133)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{137 \times 137 + 133 \times 133 + 18221}{137 \times 137 \times 137 - 133 \times 133 \times 133} \]

### Step 2: Recognize the components
Notice that \(137 \times 137\) is \(137^2\) and \(133 \times 133\) is \(133^2\). So we can rewrite the numerator:
\[ 137^2 + 133^2 + 18221 \]

### Step 3: Simplify the numerator
Next, we can calculate \(137^2\) and \(133^2\):
\[ 137^2 = 18769, \quad 133^2 = 17689 \]
Now substituting these values into the numerator:
\[ 18769 + 17689 + 18221 \]
Calculating this gives:
\[ 18769 + 17689 = 36458 \]
\[ 36458 + 18221 = 54679 \]
So, the numerator simplifies to \(54679\).

### Step 4: Simplify the denominator
Now, we simplify the denominator:
\[ 137^3 - 133^3 \]
Using the identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), where \(a = 137\) and \(b = 133\):
\[ 137^3 - 133^3 = (137 - 133)(137^2 + 137 \times 133 + 133^2) \]
Calculating \(137 - 133\):
\[ 137 - 133 = 4 \]
Now we need to calculate \(137^2 + 137 \times 133 + 133^2\):
We already have \(137^2 = 18769\) and \(133^2 = 17689\). Now we calculate \(137 \times 133\):
\[ 137 \times 133 = 18221 \]
So:
\[ 137^2 + 137 \times 133 + 133^2 = 18769 + 18221 + 17689 \]
Calculating this gives:
\[ 18769 + 18221 = 36990 \]
\[ 36990 + 17689 = 54679 \]
Thus, the denominator simplifies to:
\[ 4 \times 54679 \]

### Step 5: Final expression
Now we can write the entire expression:
\[ \frac{54679}{4 \times 54679} \]
This simplifies to:
\[ \frac{1}{4} \]

### Conclusion
Thus, the value of the expression is:
\[ \frac{1}{4} \]

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