The value of `(1)/(sqrt(7-4sqrt(3)))` is closest to

Options

A. `4.1 `
B. `4.2`
C. `1.2`
D. `3.7`

Answer

To solve the problem of finding the value of \( \frac{1}{\sqrt{7 - 4\sqrt{3}}} \), we can follow these steps:

### Step 1: Simplify the Denominator
We start with the expression:
\[ \sqrt{7 - 4\sqrt{3}} \]
We can express \( 7 \) as \( 2^2 + (\sqrt{3})^2 \) and recognize that \( 4\sqrt{3} \) can be rewritten using the formula for the square of a binomial:
\[ a^2 + b^2 - 2ab = (a - b)^2 \]
Here, \( a = 2 \) and \( b = \sqrt{3} \).

### Step 2: Rewrite the Expression
Using the identity mentioned above, we can rewrite the denominator:
\[ 7 - 4\sqrt{3} = (2 - \sqrt{3})^2 \]
Thus, we have:
\[ \sqrt{7 - 4\sqrt{3}} = \sqrt{(2 - \sqrt{3})^2} = 2 - \sqrt{3} \]

### Step 3: Substitute Back into the Original Expression
Now we can substitute back into our original expression:
\[ \frac{1}{\sqrt{7 - 4\sqrt{3}}} = \frac{1}{2 - \sqrt{3}} \]

### Step 4: Rationalize the Denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{1}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})} \]
Calculating the denominator:
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus, we have:
\[ \frac{1}{2 - \sqrt{3}} = 2 + \sqrt{3} \]

### Step 5: Approximate the Value
Now we need to approximate the value of \( 2 + \sqrt{3} \). We know that \( \sqrt{3} \approx 1.732 \):
\[ 2 + \sqrt{3} \approx 2 + 1.732 = 3.732 \]

### Conclusion
The value of \( \frac{1}{\sqrt{7 - 4\sqrt{3}}} \) is closest to \( 3.732 \).

### Final Answer
Thus, the closest value is \( \boxed{3.7} \).