Which of the following is a rational number?

Options

A. Sum of `(2+sqrt(3))` and its reciprocal
B. Square root of 18
C. Square root of `7+4sqrt(3)`
D. None of these

Answer

To determine which of the given options is a rational number, we need to analyze each option step by step.

### Step 1: Understand Rational Numbers
A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.

### Step 2: Analyze Each Option
We will evaluate each of the options provided to identify which one is a rational number.

#### Option 1: \(2 + \sqrt{3} + \frac{1}{2 + \sqrt{3}}\)
1. **Calculate the reciprocal**: The expression involves \(2 + \sqrt{3}\) and its reciprocal.
2. **Rationalize the denominator**: To simplify \(\frac{1}{2 + \sqrt{3}}\), we multiply the numerator and denominator by the conjugate \(2 - \sqrt{3}\):
\[ \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} \]
3. **Calculate the denominator**: Using the difference of squares:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
4. **Simplify**: Thus, \(\frac{1}{2 + \sqrt{3}} = 2 - \sqrt{3}\).
5. **Combine terms**: Now, we have:
\[ 2 + \sqrt{3} + (2 - \sqrt{3}) = 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \]
6. **Conclusion**: \(4\) is a rational number.

#### Option 2: \(\sqrt{18}\)
1. **Calculate**: \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).
2. **Determine rationality**: Since \(\sqrt{2}\) is irrational, \(3\sqrt{2}\) is also irrational.

#### Option 3: \(\sqrt{7} + 4\sqrt{3}\)
1. **Determine rationality**: Both \(\sqrt{7}\) and \(\sqrt{3}\) are irrational, and their sum is also irrational.

#### Option 4: (Assuming there is a fourth option)
- Since we have already established that options 2 and 3 are irrational, we conclude that option 1 is the only rational number.

### Final Answer
The rational number among the given options is **Option 1**.