1. Write whether the number \(\sqrt{9}\) is a rational number or an irrational number, with explanation.

2. Write whether the number \(\sqrt{225}\) is a rational number or an irrational number, with explanation.

3. Write whether the number \(\sqrt{50}\) is a rational number or an irrational number, with explanation.

4. Write whether the number \(\sqrt{100}\) is a rational number or an irrational number, with explanation.

5. Write whether the number \(-\sqrt{81}\) is a rational number or an irrational number, with explanation.

6. Write whether the number \(\sqrt{42}\) is a rational number or an irrational number, with explanation.

7. Write whether the number \(-\sqrt{1000}\) is a rational number or an irrational number, with explanation.

8. In a two-digit number, the unit digit is 6 more than the tens digit, and the product of the digits is 12 less than the number itself. — Form a quadratic equation with one variable from the given statement.

9. Every real number is an irrational number. This statement is **false**. While irrational numbers are a subset of real numbers, real numbers also include **rational numbers**—like integers, fractions, and terminating or repeating decimals. So not every real number is irrational. If you'd like, I can break down the number system hierarchy to make this crystal clear! ????

10. \(\sqrt{42}\) cannot be written in the form \(\dfrac{p}{q}\) (where \(p\) and \(q\) are integers and \(q \ne 0\)), so \(\sqrt{42}\) is an irrational number.

11. In a two-digit number, the unit digit is 6 more than the tens digit, and the product of the two digits is 12 less than the number itself. What could the number be?

12. Got it! Here's the cleaned-up version without the bold formatting: Determine the mode from the following frequency distribution: | Item Size Number | 4–8 | 8–12 | 12–16 | 16–20 | 20–24 | 24–28 | 28–32 | |------------------|-----|------|-------|-------|-------|-------|-------| | Frequency | 9 | 10 | 18 | 14 | 10 | 6 | 3 |

13. In a two-digit number, the unit digit is 6 more than the tens digit, and the product of the two digits is 12 less than the number itself. What are the possible values of the unit digit?

14. \(a\) is a positive number, and in the proportion \(a : \cfrac{27}{64} = \cfrac{3}{4} : a\), what is the value of \(a\)?

(a) \(\cfrac{9}{11}\) (b) \(\cfrac{3}{4}\) (c) \(\cfrac{9}{20}\) (d) \(\cfrac{9}{16}\)

15. If \(4\), \(5 + x\), \(2x + 1\), and \(14\) are in continued proportion and \(x\) is a positive number, then what is the value of \(x\)?

(a) 1 (b) 2 (c) 4 (d) 3

16. I will write a two-digit number whose digits add up to 14, and when 29 is subtracted from the number, the digits become equal. Let’s form simultaneous equations and find out what the two-digit number is.

17. If the sum and product of two quadratic factors are a rational number, then the factors are _____ factors.

18. Write the product form of a rational and an irrational number: \(\sqrt{175}\)

19. Write in the form of the product of a rational and an irrational number: \(2\sqrt{112}\)

20. Write in the form of the product of a rational and an irrational number: \(\sqrt{108}\)

21. Write in the form of the product of a rational and an irrational number: \(\sqrt{125}\)

22. Write in the form of the product of a rational and an irrational number: \(5\sqrt{119}\)

23. If the rationalizing factor of \(\sqrt{5}\) is \(\sqrt{x}\), then calculate and write the smallest possible value of \(x\), where \(x\) is a whole number.

24. A table showing how many of the 150 athletes completed the 100-meter hurdles race within different time intervals (in seconds) is given below. ```html

25. Is \(0\) a rational number? Express \(0\) in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers, \(q \ne 0\), and there is no positive common factor between \(p\) and \(q\) other than 1.

26. Addition, subtraction, multiplication, and division (when the divisor is not zero) of two rational numbers result in another rational number.

27. Every rational number is a real number. Short, true, and mathematically solid. Want to break down the hierarchy of number types next? It's more fascinating than it sounds!