Which of the following is a rational number?

Options

A. Sum of 2 + `sqrt(3)` and its inverse
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 will analyze each option step by step.

### Step 1: Understanding Rational Numbers
A rational number is defined as any number that can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Rational numbers can be terminating or repeating decimals.

### Step 2: Analyze Option A
**Option A:** \( 2 + \sqrt{3} + \frac{1}{2 + \sqrt{3}} \)

1. **Calculate the inverse:**
The inverse of \( 2 + \sqrt{3} \) is \( \frac{1}{2 + \sqrt{3}} \).

2. **Rationalize the denominator:**
To simplify \( \frac{1}{2 + \sqrt{3}} \), multiply the numerator and denominator by \( 2 - \sqrt{3} \):
\[ \frac{1 \cdot (2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3} \]

3. **Combine the terms:**
Now, we add \( 2 + \sqrt{3} \) and \( 2 - \sqrt{3} \):
\[ (2 + \sqrt{3}) + (2 - \sqrt{3}) = 2 + 2 = 4 \]

4. **Conclusion for Option A:**
Since \( 4 \) can be expressed as \( \frac{4}{1} \), it is a rational number.

### Step 3: Analyze Option B
**Option B:** \( \sqrt{18} \)

1. **Calculate the value:**
\[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \]
The value \( \sqrt{2} \) is known to be an irrational number, hence \( 3\sqrt{2} \) is also irrational.

### Step 4: Analyze Option C
**Option C:** \( \sqrt{7} + 4\sqrt{3} \)

1. **Identify the components:**
Both \( \sqrt{7} \) and \( \sqrt{3} \) are irrational numbers.

2. **Conclusion for Option C:**
The sum of two irrational numbers is not guaranteed to be rational. Therefore, \( \sqrt{7} + 4\sqrt{3} \) is also an irrational number.

### Final Conclusion
After analyzing all options:
- **Option A** is a rational number (resulting in 4).
- **Option B** is irrational (\( \sqrt{18} \)).
- **Option C** is irrational (\( \sqrt{7} + 4\sqrt{3} \)).

Thus, the answer is **Option A**: \( 2 + \sqrt{3} + \frac{1}{2 + \sqrt{3}} \) is the rational number.

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