8th Grade Common Core Math Worksheets - Free Printable
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Step-by-step solution for: 8th Grade Common Core Math Worksheets
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Step-by-step solution for: 8th Grade Common Core Math Worksheets
Here's the complete solution to all 14 problems, with clear explanations for each.
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- Rational Number: A number that can be expressed as a fraction `a/b` where `a` and `b` are integers and `b ≠ 0`. Their decimal expansions either terminate or repeat.
- Irrational Number: A number that cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating. Examples: π, √2, √3, etc.
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## ✔ Problem Solutions:
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Options:
1) π — irrational
2) 5/4 — rational (fraction of integers) ✔
3) √7 — irrational (not perfect square)
4) √(3/2) — irrational (simplifies to √6 / 2, still irrational)
> Answer: 2) 5/4
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Options:
1) √8 = 2√2 — irrational
2) π — irrational
3) 5√9 = 5 × 3 = 15 — rational ✔
4) 6√2 — irrational
> Answer: 3) 5√9
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Options:
1) π — irrational
2) √(1/2) = 1/√2 — irrational
3) √3 — irrational
4) √(1/4) = 1/2 — rational ✔
> Answer: 4) √(1/4)
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Options:
1) √9 = 3 — rational
2) 3.14 — rational (terminating decimal)
3) √3 — irrational ✔
4) 3/4 — rational
> Answer: 3) √3
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Options:
1) 0 — rational
2) π — irrational ✔
3) -1/3 — rational
4) √9 = 3 — rational
> Answer: 2) π
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This decimal does not repeat in a fixed pattern — it has increasing numbers of 1’s between 4’s → non-repeating, non-terminating.
> Answer: 3) irrational
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Options:
1) √2 — irrational ✔
2) 1/2 — rational
3) 0.17 — rational (terminating)
4) 0 — rational
> Answer: 1) √2
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Options:
1) √9 = 3 — rational
2) √8 = 2√2 — irrational ✔
3) 0.3333 — rational (repeating decimal = 1/3)
4) 2/3 — rational
> Answer: 2) √8
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Options:
1) 0.3̅ = 0.333... = 1/3 — rational
2) 3/8 — rational
3) √49 = 7 — rational
4) π — irrational ✔
> Answer: 4) π
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Options:
1) 5/4 — rational
2) 0.3̅ — rational
3) √121 = 11 — rational
4) π — irrational ✔
> Answer: 4) π
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We need:
- Real → inside square root ≥ 0 → x² − 9 ≥ 0 → |x| ≥ 3
- Irrational → result must not be integer or rational
Check options:
1) x = 5 → √(25−9) = √16 = 4 — rational ✘
2) x = 0 → √(-9) — not real ✘
3) x = -3 → √(9−9) = 0 — rational ✘
4) x = 4 → √(16−9) = √7 — irrational ✔
> Answer: 4) 4
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Given: √(4/9), √20, √121
- √(4/9) = 2/3 — rational
- √20 = √(4×5) = 2√5 — irrational ✔
- √121 = 11 — rational
> Answer: √20
Why? Because 20 is not a perfect square, and its prime factorization includes 5 (which is not squared), so √20 cannot be simplified to a rational number.
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Check each:
- √99 / 11 = √(9×11)/11 = (3√11)/11 — irrational (has √11)
- √164 = √(4×41) = 2√41 — irrational
- √196 = 14 — rational ✔
> Answer: √196
Why? 196 is a perfect square (14²), so its square root is an integer → rational.
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> Example: √2
Explanation: √2 cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating (≈1.41421356237...). This was proven by the ancient Greeks using proof by contradiction — assuming √2 = a/b leads to a contradiction if a and b are coprime integers.
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## ✔ Final Answers Summary:
1. 2) 5/4
2. 3) 5√9
3. 4) √(1/4)
4. 3) √3
5. 2) π
6. 3) irrational
7. 1) √2
8. 2) √8
9. 4) π
10. 4) π
11. 4) 4
12. √20 — because 20 is not a perfect square and simplifies to 2√5, which is irrational.
13. √196 — because 196 = 14², so √196 = 14, which is rational.
14. √2 — irrational because it cannot be written as a fraction of integers and has a non-repeating, non-terminating decimal.
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Let me know if you’d like a printable version or flashcards for review!
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Definitions Recap:
- Rational Number: A number that can be expressed as a fraction `a/b` where `a` and `b` are integers and `b ≠ 0`. Their decimal expansions either terminate or repeat.
- Irrational Number: A number that cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating. Examples: π, √2, √3, etc.
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## ✔ Problem Solutions:
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1. Which number is rational?
Options:
1) π — irrational
2) 5/4 — rational (fraction of integers) ✔
3) √7 — irrational (not perfect square)
4) √(3/2) — irrational (simplifies to √6 / 2, still irrational)
> Answer: 2) 5/4
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2. Which is a rational number?
Options:
1) √8 = 2√2 — irrational
2) π — irrational
3) 5√9 = 5 × 3 = 15 — rational ✔
4) 6√2 — irrational
> Answer: 3) 5√9
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3. Which expression is rational?
Options:
1) π — irrational
2) √(1/2) = 1/√2 — irrational
3) √3 — irrational
4) √(1/4) = 1/2 — rational ✔
> Answer: 4) √(1/4)
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4. Which is an irrational number?
Options:
1) √9 = 3 — rational
2) 3.14 — rational (terminating decimal)
3) √3 — irrational ✔
4) 3/4 — rational
> Answer: 3) √3
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5. Which is an irrational number?
Options:
1) 0 — rational
2) π — irrational ✔
3) -1/3 — rational
4) √9 = 3 — rational
> Answer: 2) π
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6. The number 0.14114111411114… is
This decimal does not repeat in a fixed pattern — it has increasing numbers of 1’s between 4’s → non-repeating, non-terminating.
> Answer: 3) irrational
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7. Which expression represents an irrational number?
Options:
1) √2 — irrational ✔
2) 1/2 — rational
3) 0.17 — rational (terminating)
4) 0 — rational
> Answer: 1) √2
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8. Which number is irrational?
Options:
1) √9 = 3 — rational
2) √8 = 2√2 — irrational ✔
3) 0.3333 — rational (repeating decimal = 1/3)
4) 2/3 — rational
> Answer: 2) √8
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9. Which is an irrational number?
Options:
1) 0.3̅ = 0.333... = 1/3 — rational
2) 3/8 — rational
3) √49 = 7 — rational
4) π — irrational ✔
> Answer: 4) π
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10. Which number is irrational?
Options:
1) 5/4 — rational
2) 0.3̅ — rational
3) √121 = 11 — rational
4) π — irrational ✔
> Answer: 4) π
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11. The value of √(x² − 9) is a real and irrational number when x is equal to
We need:
- Real → inside square root ≥ 0 → x² − 9 ≥ 0 → |x| ≥ 3
- Irrational → result must not be integer or rational
Check options:
1) x = 5 → √(25−9) = √16 = 4 — rational ✘
2) x = 0 → √(-9) — not real ✘
3) x = -3 → √(9−9) = 0 — rational ✘
4) x = 4 → √(16−9) = √7 — irrational ✔
> Answer: 4) 4
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12. Which number below is irrational?
Given: √(4/9), √20, √121
- √(4/9) = 2/3 — rational
- √20 = √(4×5) = 2√5 — irrational ✔
- √121 = 11 — rational
> Answer: √20
Why? Because 20 is not a perfect square, and its prime factorization includes 5 (which is not squared), so √20 cannot be simplified to a rational number.
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13. Given: √99 / 11, √164, √196 — Identify the rational number
Check each:
- √99 / 11 = √(9×11)/11 = (3√11)/11 — irrational (has √11)
- √164 = √(4×41) = 2√41 — irrational
- √196 = 14 — rational ✔
> Answer: √196
Why? 196 is a perfect square (14²), so its square root is an integer → rational.
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14. Write an irrational number and explain why it is irrational.
> Example: √2
Explanation: √2 cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating (≈1.41421356237...). This was proven by the ancient Greeks using proof by contradiction — assuming √2 = a/b leads to a contradiction if a and b are coprime integers.
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## ✔ Final Answers Summary:
1. 2) 5/4
2. 3) 5√9
3. 4) √(1/4)
4. 3) √3
5. 2) π
6. 3) irrational
7. 1) √2
8. 2) √8
9. 4) π
10. 4) π
11. 4) 4
12. √20 — because 20 is not a perfect square and simplifies to 2√5, which is irrational.
13. √196 — because 196 = 14², so √196 = 14, which is rational.
14. √2 — irrational because it cannot be written as a fraction of integers and has a non-repeating, non-terminating decimal.
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Let me know if you’d like a printable version or flashcards for review!
Parent Tip: Review the logic above to help your child master the concept of 8th grade math practice worksheet with answers.
