8th Grade Common Core Math Worksheets - Free Printable

Problem Analysis:

The task involves classifying numbers as rational or irrational. Here are the key definitions to keep in mind:
- Rational Numbers: Numbers that can be expressed as a ratio of two integers \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Rational numbers have decimal expansions that either terminate or repeat.
- Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers. Irrational numbers have non-terminating, non-repeating decimal expansions.

Solution to Each Question:

#### 1. Which number is rational?
Options:
1) \( \pi \)
2) \( \frac{5}{4} \)
3) \( \sqrt{7} \)
4) \( \sqrt{\frac{3}{2}} \)

- \( \pi \): Irrational (non-terminating, non-repeating decimal).
- \( \frac{5}{4} \): Rational (it is a fraction of two integers).
- \( \sqrt{7} \): Irrational (square root of a non-perfect square).
- \( \sqrt{\frac{3}{2}} \): Irrational (square root of a fraction that is not a perfect square).

Answer: 2) \( \frac{5}{4} \)

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#### 2. Which is a rational number?
Options:
1) \( \sqrt{8} \)
2) \( \pi \)
3) \( 5\sqrt{9} \)
4) \( 6\sqrt{2} \)

- \( \sqrt{8} \): Irrational (square root of a non-perfect square).
- \( \pi \): Irrational (well-known irrational number).
- \( 5\sqrt{9} \): Rational (\( \sqrt{9} = 3 \), so \( 5\sqrt{9} = 5 \times 3 = 15 \)).
- \( 6\sqrt{2} \): Irrational (multiplying an integer by the square root of a non-perfect square results in an irrational number).

Answer: 3) \( 5\sqrt{9} \)

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#### 3. Which expression is rational?
Options:
1) \( \pi \)
2) \( \sqrt{\frac{1}{2}} \)
3) \( \sqrt{3} \)
4) \( \sqrt{\frac{1}{4}} \)

- \( \pi \): Irrational.
- \( \sqrt{\frac{1}{2}} \): Irrational (square root of a fraction that is not a perfect square).
- \( \sqrt{3} \): Irrational (square root of a non-perfect square).
- \( \sqrt{\frac{1}{4}} \): Rational (\( \sqrt{\frac{1}{4}} = \frac{1}{2} \)).

Answer: 4) \( \sqrt{\frac{1}{4}} \)

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#### 4. Which is an irrational number?
Options:
1) \( \sqrt{9} \)
2) \( 3.14 \)
3) \( \sqrt{3} \)
4) \( \frac{3}{4} \)

- \( \sqrt{9} \): Rational (\( \sqrt{9} = 3 \)).
- \( 3.14 \): Rational (terminating decimal).
- \( \sqrt{3} \): Irrational (square root of a non-perfect square).
- \( \frac{3}{4} \): Rational (fraction of two integers).

Answer: 3) \( \sqrt{3} \)

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#### 5. Which is an irrational number?
Options:
1) \( 0 \)
2) \( \pi \)
3) \( -\frac{1}{3} \)
4) \( \sqrt{9} \)

- \( 0 \): Rational (integer).
- \( \pi \): Irrational.
- \( -\frac{1}{3} \): Rational (fraction of two integers).
- \( \sqrt{9} \): Rational (\( \sqrt{9} = 3 \)).

Answer: 2) \( \pi \)

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#### 6. The number \( 0.14114111411114 \ldots \) is
Options:
1) integral
2) rational
3) irrational
4) whole

- The number \( 0.14114111411114 \ldots \) has a non-terminating, non-repeating decimal expansion. Therefore, it is irrational.

Answer: 3) irrational

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#### 7. Which expression represents an irrational number?
Options:
1) \( \sqrt{2} \)
2) \( \frac{1}{2} \)
3) \( 0.17 \)
4) \( 0 \)

- \( \sqrt{2} \): Irrational (square root of a non-perfect square).
- \( \frac{1}{2} \): Rational (fraction of two integers).
- \( 0.17 \): Rational (terminating decimal).
- \( 0 \): Rational (integer).

Answer: 1) \( \sqrt{2} \)

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#### 8. Which number is irrational?
Options:
1) \( \sqrt{9} \)
2) \( \sqrt{8} \)
3) \( 0.3333 \)
4) \( \frac{2}{3} \)

- \( \sqrt{9} \): Rational (\( \sqrt{9} = 3 \)).
- \( \sqrt{8} \): Irrational (square root of a non-perfect square).
- \( 0.3333 \): Rational (repeating decimal).
- \( \frac{2}{3} \): Rational (fraction of two integers).

Answer: 2) \( \sqrt{8} \)

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#### 9. Which is an irrational number?
Options:
1) \( 0.\overline{3} \)
2) \( \frac{3}{8} \)
3) \( \sqrt{49} \)
4) \( \pi \)

- \( 0.\overline{3} \): Rational (repeating decimal).
- \( \frac{3}{8} \): Rational (fraction of two integers).
- \( \sqrt{49} \): Rational (\( \sqrt{49} = 7 \)).
- \( \pi \): Irrational.

Answer: 4) \( \pi \)

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#### 10. Which number is irrational?
Options:
1) \( \frac{5}{4} \)
2) \( 0.\overline{3} \)
3) \( \sqrt{121} \)
4) \( \pi \)

- \( \frac{5}{4} \): Rational (fraction of two integers).
- \( 0.\overline{3} \): Rational (repeating decimal).
- \( \sqrt{121} \): Rational (\( \sqrt{121} = 11 \)).
- \( \pi \): Irrational.

Answer: 4) \( \pi \)

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#### 11. The value of \( \sqrt{x^2 - 9} \) is a real and irrational number when \( x \) is equal to
Options:
1) 5
2) 0
3) -3
4) 4

- For \( x = 5 \): \( \sqrt{5^2 - 9} = \sqrt{25 - 9} = \sqrt{16} = 4 \) (rational).
- For \( x = 0 \): \( \sqrt{0^2 - 9} = \sqrt{-9} \) (not real).
- For \( x = -3 \): \( \sqrt{(-3)^2 - 9} = \sqrt{9 - 9} = \sqrt{0} = 0 \) (rational).
- For \( x = 4 \): \( \sqrt{4^2 - 9} = \sqrt{16 - 9} = \sqrt{7} \) (irrational).

Answer: 4) 4

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#### 12. Which number below is irrational?
Numbers:
- \( \sqrt{\frac{4}{9}} \)
- \( \sqrt{20} \)
- \( \sqrt{121} \)

- \( \sqrt{\frac{4}{9}} \): Rational (\( \sqrt{\frac{4}{9}} = \frac{2}{3} \)).
- \( \sqrt{20} \): Irrational (square root of a non-perfect square).
- \( \sqrt{121} \): Rational (\( \sqrt{121} = 11 \)).

Answer: \( \sqrt{20} \)

Why is it irrational? Because \( \sqrt{20} \) simplifies to \( 2\sqrt{5} \), and \( \sqrt{5} \) is irrational.

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#### 13. Given: \( \frac{\sqrt{99}}{11}, \sqrt{64}, \sqrt{196} \). Identify the expression that is a rational number and explain why it is rational.

- \( \frac{\sqrt{99}}{11} \): Irrational (\( \sqrt{99} \) is irrational, and dividing it by 11 does not make it rational).
- \( \sqrt{64} \): Rational (\( \sqrt{64} = 8 \)).
- \( \sqrt{196} \): Rational (\( \sqrt{196} = 14 \)).

Both \( \sqrt{64} \) and \( \sqrt{196} \) are rational. However, if only one answer is expected, we can choose either. For clarity, let's choose \( \sqrt{64} \).

Answer: \( \sqrt{64} \)

Explanation: \( \sqrt{64} = 8 \), which is an integer and therefore rational.

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#### 14. Write an irrational number and explain why it is irrational.

An example of an irrational number is \( \sqrt{2} \).

Explanation: \( \sqrt{2} \) is irrational because it cannot be expressed as a ratio of two integers, and its decimal expansion is non-terminating and non-repeating.

Answer: \( \sqrt{2} \)

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Final Answers:

1. \( \boxed{2} \)
2. \( \boxed{3} \)
3. \( \boxed{4} \)
4. \( \boxed{3} \)
5. \( \boxed{2} \)
6. \( \boxed{3} \)
7. \( \boxed{1} \)
8. \( \boxed{2} \)
9. \( \boxed{4} \)
10. \( \boxed{4} \)
11. \( \boxed{4} \)
12. \( \boxed{\sqrt{20}} \)
13. \( \boxed{\sqrt{64}} \)
14. \( \boxed{\sqrt{2}} \)