What would be the result of the following expression
`((111+189)^2 - (189-111)^2)/(2 xx 111xx189)`

Options

A. 78
B. 300
C. 2
D. 4

Answer

To solve the expression \(\frac{(111 + 189)^2 - (189 - 111)^2}{2 \times 111 \times 189}\), we can follow these steps:

### Step 1: Identify the terms
Let \( a = 111 + 189 \) and \( b = 189 - 111 \).

### Step 2: Calculate \( a \) and \( b \)
- \( a = 111 + 189 = 300 \)
- \( b = 189 - 111 = 78 \)

### Step 3: Apply the difference of squares formula
The difference of squares formula states that \( x^2 - y^2 = (x + y)(x - y) \). Here, we can rewrite the expression as:
\[ \frac{(a^2 - b^2)}{2 \times 111 \times 189} = \frac{(a + b)(a - b)}{2 \times 111 \times 189} \]

### Step 4: Calculate \( a + b \) and \( a - b \)
- \( a + b = 300 + 78 = 378 \)
- \( a - b = 300 - 78 = 222 \)

### Step 5: Substitute back into the expression
Now, substituting \( a + b \) and \( a - b \) into the expression gives:
\[ \frac{(378)(222)}{2 \times 111 \times 189} \]

### Step 6: Calculate the denominator
Calculate \( 2 \times 111 \times 189 \):
\[ 2 \times 111 \times 189 = 2 \times 21099 = 42198 \]

### Step 7: Calculate the numerator
Calculate \( 378 \times 222 \):
\[ 378 \times 222 = 84036 \]

### Step 8: Divide the numerator by the denominator
Now we divide:
\[ \frac{84036}{42198} = 2 \]

### Final Answer
Thus, the result of the expression is \( 2 \).

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