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The image contains a multiple-choice question. The question asks the user to identify statements related to rational numbers, specifically focused on identifying rational numbers based on their decimal representation or their relationship to perfect squares.

Here are the options given:

1. \( \sqrt{2} \) is rational because it is a repeating decimal.
2. \( \sqrt{81} \) is rational because 81 is a perfect square.
3. \( \frac{7}{24} \) is rational because it is a fraction involving non-repeating, non-terminating decimals.
4. \( 5.676767... \) is rational because it is a repeating decimal.

### Analyzing each statement:
1. \( \sqrt{2} \) is *irrational*, as it cannot be expressed as a fraction and has a non-repeating, non-terminating decimal expansion.
2. \( \sqrt{81} = 9 \), and since 9 is an integer, this is a rational number. So, this statement is **true**.
3. \( \frac{7}{24} \) is a rational number because it's a fraction of two integers, but it doesn't have a non-repeating, non-terminating decimal expansion (the decimal expansion is terminating), so the reasoning in the statement is **false**.
4. \( 5.676767... \) is a repeating decimal, and repeating decimals are rational numbers, so this statement is **true**.

### Conclusion:
The correct statements are:
- \( \sqrt{81} \) is rational because 81 is a perfect square.
- \( 5.676767... \) is rational because it is a repeating decimal.

Would you like a more detailed explanation of rational vs. irrational numbers?

### 5 Related Questions:
1. What is the difference between repeating and non-repeating decimals?
2. How can we prove that \( \sqrt{2} \) is irrational?
3. Can all perfect squares be classified as rational numbers?
4. What are examples of other irrational numbers besides \( \sqrt{2} \)?
5. How can you convert repeating decimals into fractions?

### Tip:
To determine if a number is rational, check if its decimal expansion either terminates or repeats!

Select all the rational numbers from the list: (1) √2 is rational because it is a repeating decimal. (2) √81 is rational because 81 is a perfect square. (3) 7/24 is rational because it is a fraction involving non-repeating, non-terminating decimals. (4) 5.676767... is rational because it is a repeating decimal.

Learn how to identify rational numbers by analyzing perfect squares, fractions, and repeating decimals. This exercise covers key concepts in rational and irrational numbers, explaining the reasoning behind common misconceptions like √2 and √81. Suitable for students in Grades 8-10.

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