Check which of the following numbers is rational .

Options

A. `sqrt(9/27)`
B. `sqrt196`
C. `(1 + sqrt5)+(4+sqrt5)`
D. 1.010010001...

Explanation

To determine which of the following numbers is rational, we will analyze each number one by one.

1. **First Number: \(\sqrt{\frac{9}{27}}\)**

Step 1: Simplify the fraction \(\frac{9}{27}\).
\[ \frac{9}{27} = \frac{1}{3} \]

Step 2: Take the square root of \(\frac{1}{3}\).
\[ \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]
Since \(\sqrt{3}\) is an irrational number, \(\frac{1}{\sqrt{3}}\) is also irrational.

**Conclusion**: The first number is irrational.

2. **Second Number: \(\sqrt{196}\)**

Step 1: Calculate the square root of 196.
\[ \sqrt{196} = 14 \]
Since 14 is a whole number, it is a rational number.

**Conclusion**: The second number is rational.

3. **Third Number: \(1 + \sqrt{5} + 4 + \sqrt{5}\)**

Step 1: Combine like terms.
\[ 1 + 4 + \sqrt{5} + \sqrt{5} = 5 + 2\sqrt{5} \]
Since \(\sqrt{5}\) is irrational, \(2\sqrt{5}\) is also irrational, and thus \(5 + 2\sqrt{5}\) is irrational.

**Conclusion**: The third number is irrational.

4. **Fourth Number: \(1.010010001...\)**

Step 1: Analyze the decimal.
This number is non-terminating and non-repeating.

**Conclusion**: The fourth number is irrational.

### Final Summary:
- The first number is irrational.
- The second number is rational.
- The third number is irrational.
- The fourth number is irrational.