`((998)^(2) - (997)^(2) - 45)/((98)^(2) - (97)^(2))` equals

Options

A. 1995
B. 195
C. 95
D. 10

Answer

To solve the expression \(\frac{(998)^2 - (997)^2 - 45}{(98)^2 - (97)^2}\), we can use the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\).

### Step-by-Step Solution:

1. **Identify the Numerator:**
The numerator is \((998)^2 - (997)^2 - 45\).
We can apply the difference of squares formula:
\[ (998)^2 - (997)^2 = (998 + 997)(998 - 997) \]
This simplifies to:
\[ (1995)(1) = 1995 \]
Therefore, the numerator becomes:
\[ 1995 - 45 = 1950 \]

2. **Identify the Denominator:**
The denominator is \((98)^2 - (97)^2\).
Again, using the difference of squares formula:
\[ (98)^2 - (97)^2 = (98 + 97)(98 - 97) \]
This simplifies to:
\[ (195)(1) = 195 \]

3. **Combine the Results:**
Now we can substitute back into the original expression:
\[ \frac{1950}{195} \]

4. **Simplify the Fraction:**
We can simplify the fraction:
\[ \frac{1950 \div 195}{195 \div 195} = \frac{10}{1} = 10 \]

Thus, the final answer is:
\[ \boxed{10} \]