What would be the result of the following expression `((971 +29)^(2) - (1971-29)^2)/(2 xx 971 xx 29)`

Options

A. 2
B. 1
C. 8
D. 4

Answer

To solve the expression \(\frac{(971 + 29)^2 - (971 - 29)^2}{2 \times 971 \times 29}\), we can follow these steps:

### Step 1: Identify the expression
The expression we need to simplify is:
\[ \frac{(971 + 29)^2 - (971 - 29)^2}{2 \times 971 \times 29} \]

### Step 2: Apply the difference of squares formula
Recall that \(a^2 - b^2 = (a + b)(a - b)\). Here, we can let:
- \(a = 971 + 29\)
- \(b = 971 - 29\)

Thus, we can rewrite the numerator:
\[ (971 + 29)^2 - (971 - 29)^2 = [(971 + 29) + (971 - 29)][(971 + 29) - (971 - 29)] \]

### Step 3: Simplify the terms
Calculating the two parts:
1. \((971 + 29) + (971 - 29) = 971 + 29 + 971 - 29 = 2 \times 971\)
2. \((971 + 29) - (971 - 29) = 971 + 29 - 971 + 29 = 2 \times 29\)

So, we can substitute these back into our expression:
\[ = \frac{(2 \times 971)(2 \times 29)}{2 \times 971 \times 29} \]

### Step 4: Cancel out common terms
Now, we can simplify:
\[ = \frac{4 \times 971 \times 29}{2 \times 971 \times 29} \]
Cancelling \(2 \times 971 \times 29\) from the numerator and denominator gives:
\[ = 2 \]

### Final Result
Thus, the result of the expression is:
\[ \boxed{2} \]