If `m=7+4 sqrt(3),(sqrt(m)+(1)/(sqrt(m)))=?` / यदि ` m = 7 + 4 sqrt(3) , ( sqrt( m) + (1)/( sqrtm ) ) = ?`

Options

A. -4
B. `-2 sqrt(3)`
C. `2 sqrt(3)`
D. `4`

Explanation

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

### Step-by-Step Solution:

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

2. **Find \( \sqrt{m} \)**:
To find \( \sqrt{m} \), we can express \( m \) in a form that allows us to take the square root easily. Notice that:
\[ m = (2 + \sqrt{3})^2 \]
This is because:
\[ (2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]
Therefore, we have:
\[ \sqrt{m} = 2 + \sqrt{3} \]

3. **Calculate \( \frac{1}{\sqrt{m}} \)**:
Now we need to find \( \frac{1}{\sqrt{m}} \):
\[ \frac{1}{\sqrt{m}} = \frac{1}{2 + \sqrt{3}} \]
To simplify this, we can rationalize the denominator:
\[ \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3} \]

4. **Combine \( \sqrt{m} \) and \( \frac{1}{\sqrt{m}} \)**:
Now we can find \( \sqrt{m} + \frac{1}{\sqrt{m}} \):
\[ \sqrt{m} + \frac{1}{\sqrt{m}} = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \]

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