`sqrt({(65)^2-(16)^2}) = ?`

Options

A. 43
B. 47
C. 63
D. 67

Answer

To solve the expression \( \sqrt{(65)^2 - (16)^2} \), we can use the difference of squares identity, which states that:

\[ a^2 - b^2 = (a + b)(a - b) \]

Here, we identify \( a = 65 \) and \( b = 16 \).

### Step-by-Step Solution:

1. **Identify \( a \) and \( b \)**:
\[ a = 65, \quad b = 16 \]

2. **Apply the difference of squares formula**:
\[ (65)^2 - (16)^2 = (65 + 16)(65 - 16) \]

3. **Calculate \( 65 + 16 \)**:
\[ 65 + 16 = 81 \]

4. **Calculate \( 65 - 16 \)**:
\[ 65 - 16 = 49 \]

5. **Substitute back into the equation**:
\[ (65)^2 - (16)^2 = 81 \times 49 \]

6. **Take the square root**:
\[ \sqrt{(65)^2 - (16)^2} = \sqrt{81 \times 49} \]

7. **Simplify the square root**:
\[ \sqrt{81} \times \sqrt{49} = 9 \times 7 \]

8. **Calculate the final result**:
\[ 9 \times 7 = 63 \]

Thus, the final answer is:
\[ \sqrt{(65)^2 - (16)^2} = 63 \]