If `x = 7 - 4 sqrt(3)` , then `sqrt(x) + (1)/(sqrt(x))` is equal to

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem where \( x = 7 - 4\sqrt{3} \) and we need to find \( \sqrt{x} + \frac{1}{\sqrt{x}} \), we can follow these steps:

### Step 1: Identify the value of \( x \)
We are given:
\[ x = 7 - 4\sqrt{3} \]

### Step 2: Express \( x \) in a different form
Notice that \( 7 - 4\sqrt{3} \) can be rewritten as:
\[ x = 7 - 2(2\sqrt{3}) \]
This suggests a potential perfect square form.

### Step 3: Recognize the perfect square
We can express \( x \) as:
\[ x = (2 - \sqrt{3})^2 \]
This is because:
\[ (2 - \sqrt{3})^2 = 2^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

### Step 4: Find \( \sqrt{x} \)
Taking the square root of both sides, we have:
\[ \sqrt{x} = 2 - \sqrt{3} \]

### Step 5: Find \( \frac{1}{\sqrt{x}} \)
Now, we need to find \( \frac{1}{\sqrt{x}} \):
\[ \frac{1}{\sqrt{x}} = \frac{1}{2 - \sqrt{3}} \]
To rationalize the denominator, multiply the numerator and denominator by \( 2 + \sqrt{3} \):
\[ \frac{1}{\sqrt{x}} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})} = \frac{2 + \sqrt{3}}{4 - 3} = 2 + \sqrt{3} \]

### Step 6: Add \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \)
Now we can find \( \sqrt{x} + \frac{1}{\sqrt{x}} \):
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = (2 - \sqrt{3}) + (2 + \sqrt{3}) \]
The \( -\sqrt{3} \) and \( +\sqrt{3} \) cancel out:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = 2 + 2 = 4 \]

### Final Answer
Thus, the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \) is:
\[ \boxed{4} \]