Find the value of `((523+265)^2-(523-265)^2)/(523xx265)`

Options

A. 788
B. 2
C. 4
D. 258

Answer

To solve the expression \(\frac{(523 + 265)^2 - (523 - 265)^2}{523 \times 265}\), we can use the difference of squares formula. Let's break it down step by step.

### Step 1: Recognize the difference of squares
The expression \((a^2 - b^2)\) can be factored using the identity:
\[ a^2 - b^2 = (a - b)(a + b) \]
In our case, let:
- \(a = (523 + 265)\)
- \(b = (523 - 265)\)

### Step 2: Apply the identity
Now, we can rewrite the numerator:
\[ (523 + 265)^2 - (523 - 265)^2 = [(523 + 265) - (523 - 265)][(523 + 265) + (523 - 265)] \]

### Step 3: Simplify the terms
Calculating the first part:
\[ (523 + 265) - (523 - 265) = 523 + 265 - 523 + 265 = 2 \times 265 = 530 \]
Now for the second part:
\[ (523 + 265) + (523 - 265) = 523 + 265 + 523 - 265 = 2 \times 523 = 1046 \]

### Step 4: Combine the results
Now we can substitute back into the expression:
\[ (530)(1046) \]

### Step 5: Calculate the numerator
Thus, the numerator becomes:
\[ 530 \times 1046 \]

### Step 6: Substitute back into the original expression
Now we can substitute this back into the original expression:
\[ \frac{(530)(1046)}{523 \times 265} \]

### Step 7: Simplify the expression
Notice that:
\[ 530 = 2 \times 265 \]
So we can rewrite the numerator:
\[ \frac{(2 \times 265)(1046)}{523 \times 265} \]
The \(265\) in the numerator and denominator cancels out:
\[ = \frac{2 \times 1046}{523} \]

### Step 8: Calculate the final value
Now we can calculate:
\[ = 2 \times 2 = 4 \]

Thus, the final value is:
\[ \boxed{4} \]