If `m=7-4sqrt3,"then "(sqrtm+1/sqrtm)=?`

Options

A. 8
B. 3
C. 4
D. 9

Answer

To solve the problem, we need to find the value of \( \sqrt{m} + \frac{1}{\sqrt{m}} \) given \( m = 7 - 4\sqrt{3} \).

### Step-by-Step Solution:

1. **Substitute the value of \( m \)**:
\[ \sqrt{m} + \frac{1}{\sqrt{m}} = \sqrt{7 - 4\sqrt{3}} + \frac{1}{\sqrt{7 - 4\sqrt{3}}} \]

2. **Let \( x = \sqrt{m} \)**:
\[ x = \sqrt{7 - 4\sqrt{3}} \]
Then we need to find \( x + \frac{1}{x} \).

3. **Square both sides**:
\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]

4. **Find \( x^2 \)**:
\[ x^2 = 7 - 4\sqrt{3} \]

5. **Find \( \frac{1}{x^2} \)**:
\[ \frac{1}{x^2} = \frac{1}{7 - 4\sqrt{3}} \]
To simplify this, we can rationalize the denominator:
\[ \frac{1}{7 - 4\sqrt{3}} \cdot \frac{7 + 4\sqrt{3}}{7 + 4\sqrt{3}} = \frac{7 + 4\sqrt{3}}{(7 - 4\sqrt{3})(7 + 4\sqrt{3})} \]
The denominator simplifies to:
\[ 7^2 - (4\sqrt{3})^2 = 49 - 48 = 1 \]
Thus,
\[ \frac{1}{x^2} = 7 + 4\sqrt{3} \]

6. **Combine \( x^2 \) and \( \frac{1}{x^2} \)**:
\[ x^2 + \frac{1}{x^2} = (7 - 4\sqrt{3}) + (7 + 4\sqrt{3}) = 14 \]

7. **Use the squared equation**:
\[ \left( x + \frac{1}{x} \right)^2 = 14 + 2 = 16 \]

8. **Take the square root**:
\[ x + \frac{1}{x} = \sqrt{16} = 4 \quad \text{or} \quad -4 \]

### Final Answer:
\[ \sqrt{m} + \frac{1}{\sqrt{m}} = 4 \quad \text{or} \quad -4 \]