Worksheet for matching rational and irrational number operations with their results.

Problem Analysis:

The task involves matching mathematical expressions (multiplication and addition of rational and irrational numbers) to their results, which are either rational or irrational. The goal is to determine whether the result of each operation is a rational or irrational number.

#### Key Definitions:
1. Rational Numbers: Numbers that can be expressed as the ratio of two integers \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Examples: \( \frac{1}{2}, 3, -4.5 \).
2. Irrational Numbers: Numbers that cannot be expressed as the ratio of two integers. They have non-repeating, non-terminating decimal expansions. Examples: \( \sqrt{2}, \pi, e \).

#### Rules for Rational/Irrational Results:
- Multiplication/Division:
- Rational × Rational = Rational
- Rational × Irrational = Irrational
- Irrational × Irrational = Can be Rational or Irrational (e.g., \( \sqrt{2} \times \sqrt{2} = 2 \))
- Addition/Subtraction:
- Rational + Rational = Rational
- Rational + Irrational = Irrational
- Irrational + Irrational = Can be Rational or Irrational (e.g., \( \sqrt{2} + (-\sqrt{2}) = 0 \))

Step-by-Step Solution:

#### 1. \( 22 + 0.02 \)
- Both \( 22 \) and \( 0.02 \) are rational numbers.
- Sum of two rational numbers is rational.
- Result: Rational
- Match: a. \( \frac{469}{20} \) (Rational)

#### 2. \( 37 + 0.04 \)
- Both \( 37 \) and \( 0.04 \) are rational numbers.
- Sum of two rational numbers is rational.
- Result: Rational
- Match: g. \( 13.6055 \) (Rational)

#### 3. \( \sqrt{21} + 2 \)
- \( \sqrt{21} \) is irrational, and \( 2 \) is rational.
- Sum of an irrational number and a rational number is irrational.
- Result: Irrational
- Match: d. \( 20.48 \) (Irrational)

#### 4. \( \sqrt{13} + 10 \)
- \( \sqrt{13} \) is irrational, and \( 10 \) is rational.
- Sum of an irrational number and a rational number is irrational.
- Result: Irrational
- Match: f. \( \frac{304}{3} \) (Irrational)

#### 5. \( \sqrt{41} + 14 \)
- \( \sqrt{41} \) is irrational, and \( 14 \) is rational.
- Sum of an irrational number and a rational number is irrational.
- Result: Irrational
- Match: j. \( 52.31 \) (Irrational)

#### 6. \( \sqrt{19} \times 12 \)
- \( \sqrt{19} \) is irrational, and \( 12 \) is rational.
- Product of an irrational number and a rational number is irrational.
- Result: Irrational
- Match: c. \( 76.49 \) (Irrational)

#### 7. \( \sqrt{28} \times 15 \)
- \( \sqrt{28} \) is irrational (since \( \sqrt{28} = 2\sqrt{7} \)), and \( 15 \) is rational.
- Product of an irrational number and a rational number is irrational.
- Result: Irrational
- Match: b. \( 6.5825 \) (Irrational)

#### 8. \( \sqrt{1.64} \times \sqrt{2.89} \)
- \( \sqrt{1.64} = \sqrt{\frac{164}{100}} = \frac{\sqrt{164}}{10} \), and \( \sqrt{2.89} = \sqrt{\frac{289}{100}} = \frac{\sqrt{289}}{10} = \frac{17}{10} \).
- \( \sqrt{164} \) is irrational, but the product simplifies as:
\[
\sqrt{1.64} \times \sqrt{2.89} = \sqrt{1.64 \times 2.89} = \sqrt{4.7356} = 2.176
\]
- Result is a rational number (since \( 2.176 = \frac{2176}{1000} \)).
- Result: Rational
- Match: h. \( \frac{37}{1} \) (Rational)

#### 9. \( \sqrt{256} \times \sqrt{361} \)
- \( \sqrt{256} = 16 \) (rational), and \( \sqrt{361} = 19 \) (rational).
- Product of two rational numbers is rational.
- \( 16 \times 19 = 304 \).
- Result: Rational
- Match: i. \( \frac{304}{1} \) (Rational)

#### 10. \( \sqrt{529} \times \sqrt{400} \)
- \( \sqrt{529} = 23 \) (rational), and \( \sqrt{400} = 20 \) (rational).
- Product of two rational numbers is rational.
- \( 23 \times 20 = 460 \).
- Result: Rational
- Match: e. \( \frac{23}{1} \) (Rational)

Final Matches:

1. \( 22 + 0.02 \) → a. \( \frac{469}{20} \) (Rational)
2. \( 37 + 0.04 \) → g. \( 13.6055 \) (Rational)
3. \( \sqrt{21} + 2 \) → d. \( 20.48 \) (Irrational)
4. \( \sqrt{13} + 10 \) → f. \( \frac{304}{3} \) (Irrational)
5. \( \sqrt{41} + 14 \) → j. \( 52.31 \) (Irrational)
6. \( \sqrt{19} \times 12 \) → c. \( 76.49 \) (Irrational)
7. \( \sqrt{28} \times 15 \) → b. \( 6.5825 \) (Irrational)
8. \( \sqrt{1.64} \times \sqrt{2.89} \) → h. \( \frac{37}{1} \) (Rational)
9. \( \sqrt{256} \times \sqrt{361} \) → i. \( \frac{304}{1} \) (Rational)
10. \( \sqrt{529} \times \sqrt{400} \) → e. \( \frac{23}{1} \) (Rational)

Final Answer:

\[
\boxed{
1. a, 2. g, 3. d, 4. f, 5. j, 6. c, 7. b, 8. h, 9. i, 10. e
}
\]