The value of `sqrt(117^(2)-108^(2))`

Options

A. 55
B. 45
C. 35
D. 65

Explanation

To solve the expression \( \sqrt{117^2 - 108^2} \), 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 A and B**:
Here, we can let \( A = 117 \) and \( B = 108 \).

2. **Apply the Difference of Squares Formula**:
Using the formula \( A^2 - B^2 = (A + B)(A - B) \), we can rewrite the expression:
\[ 117^2 - 108^2 = (117 + 108)(117 - 108) \]

3. **Calculate \( A + B \) and \( A - B \)**:
- Calculate \( A + B \):
\[ 117 + 108 = 225 \]
- Calculate \( A - B \):
\[ 117 - 108 = 9 \]

4. **Substitute Back into the Expression**:
Now substitute these values back into the expression:
\[ \sqrt{117^2 - 108^2} = \sqrt{(225)(9)} \]

5. **Calculate the Product**:
Calculate the product inside the square root:
\[ 225 \times 9 = 2025 \]

6. **Find the Square Root**:
Now, we find the square root of 2025:
\[ \sqrt{2025} = 45 \]

### Final Answer:
Thus, the value of \( \sqrt{117^2 - 108^2} \) is \( 45 \).

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