If m = 9 and `n = (1)/(3)m`, then `sqrt((m)^(2) - (n)^(2) = ?)`

Options

A. `2sqrt(2)`
B. `6sqrt(2)`
C. `4sqrt(2)`
D. `5 sqrt(2)`

Answer

To solve the problem step by step, we start with the given values:

1. **Given Values**:
- \( m = 9 \)
- \( n = \frac{1}{3}m \)

2. **Calculate \( n \)**:
- Substitute \( m \) into the equation for \( n \):
\[ n = \frac{1}{3} \times 9 = 3 \]

3. **Set Up the Expression**:
- We need to find \( \sqrt{m^2 - n^2} \).
- Substitute the values of \( m \) and \( n \):
\[ \sqrt{m^2 - n^2} = \sqrt{9^2 - 3^2} \]

4. **Calculate \( m^2 \) and \( n^2 \)**:
- Calculate \( m^2 \):
\[ m^2 = 9^2 = 81 \]
- Calculate \( n^2 \):
\[ n^2 = 3^2 = 9 \]

5. **Substitute Back into the Expression**:
- Now substitute \( m^2 \) and \( n^2 \) into the square root expression:
\[ \sqrt{81 - 9} \]

6. **Perform the Subtraction**:
- Calculate \( 81 - 9 \):
\[ 81 - 9 = 72 \]

7. **Calculate the Square Root**:
- Now we need to find \( \sqrt{72} \).
- Factor \( 72 \):
\[ 72 = 36 \times 2 \]
- Taking the square root:
\[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \]

8. **Final Answer**:
- Therefore, the final answer is:
\[ \sqrt{m^2 - n^2} = 6\sqrt{2} \]