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Real Numbers and Irrational Numbers Practice Tests

These Grade 8 Math practice tests cover real numbers, rational and irrational numbers, number lines, square roots, cube roots, exact radical forms, comparisons, approximations, and applications of the Pythagorean theorem.

Skills covered

  • Classifying natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers
  • Estimating and comparing square roots and cube roots using perfect powers and number-line reasoning
  • Simplifying radical expressions and preserving exact values with LaTeX notation
  • Using real numbers in distance, area, diagonal, circumference, and right-triangle problems
  • Distinguishing exact forms from rounded decimal approximations

How to use these practice tests

Classify each number first, then choose an exact form or estimate as requested. When comparing radicals, use nearby perfect squares or a calculator estimate and check the sign.

For more Grade 8 Math practice, return to the Grade 8 Math category.

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Grade 8 Real Numbers and Irrational Numbers Practice Tests

Practice Grade 8 Math questions about real numbers, rational and irrational numbers, square roots, cube roots, exact forms, comparisons, and approximations.

Choose the best answer for each question. Use exact radical forms when requested and review the explanation after submitting your answers.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

What is the value of 25\sqrt{25}?

Please select an answer first.

Correct answer: A

The correct answer is 55.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 5⋅5=255\cdot5=25, and the principal square root is the nonnegative value.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

5⋅5=255\cdot5=25, and the principal square root is the nonnegative value.

2. QuestionMultiple choice

What is the value of 36\sqrt{36}?

Please select an answer first.

Correct answer: B

The correct answer is 66.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 6⋅6=366\cdot6=36, so the principal square root of 36 is 6.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

6⋅6=366\cdot6=36, so the principal square root of 36 is 6.

3. QuestionMultiple choice

Between which two consecutive integers is 50\sqrt{50}?

Please select an answer first.

Correct answer: C

The correct answer is Between 7 and 8.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: Because 72=497^2=49 and 82=648^2=64, 50\sqrt{50} lies between 7 and 8.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

Because 72=497^2=49 and 82=648^2=64, 50\sqrt{50} lies between 7 and 8.

4. QuestionMultiple choice

Between which two consecutive integers is 90\sqrt{90}?

Please select an answer first.

Correct answer: D

The correct answer is Between 9 and 10.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: Because 92=819^2=81 and 102=10010^2=100, 90\sqrt{90} lies between 9 and 10.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

Because 92=819^2=81 and 102=10010^2=100, 90\sqrt{90} lies between 9 and 10.

5. QuestionMultiple choice

Which is the best estimate for 10\sqrt{10}?

Please select an answer first.

Correct answer: A

The correct answer is 3.163.16.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 3.162=9.98563.16^2=9.9856, which is very close to 10, so 3.16 is a reasonable estimate.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

3.162=9.98563.16^2=9.9856, which is very close to 10, so 3.16 is a reasonable estimate.

6. QuestionMultiple choice

Which is the simplified form of 72\sqrt{72}?

Please select an answer first.

Correct answer: B

The correct answer is 626\sqrt{2}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 72=36⋅2=362=62\sqrt{72}=\sqrt{36\cdot2}=\sqrt{36}\sqrt{2}=6\sqrt{2}.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

72=36⋅2=362=62\sqrt{72}=\sqrt{36\cdot2}=\sqrt{36}\sqrt{2}=6\sqrt{2}.

7. QuestionMultiple choice

Which is the simplified form of 12\sqrt{12}?

Please select an answer first.

Correct answer: C

The correct answer is 232\sqrt{3}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 12=4⋅3=43=23\sqrt{12}=\sqrt{4\cdot3}=\sqrt{4}\sqrt{3}=2\sqrt{3}.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

12=4⋅3=43=23\sqrt{12}=\sqrt{4\cdot3}=\sqrt{4}\sqrt{3}=2\sqrt{3}.

8. QuestionMultiple choice

Which comparison is true?

Please select an answer first.

Correct answer: D

The correct answer is 5>2.2\sqrt{5}>2.2.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 5≈2.236\sqrt{5}\approx2.236, so it is greater than 2.2.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

5≈2.236\sqrt{5}\approx2.236, so it is greater than 2.2.

9. QuestionMultiple choice

What does the principal square root of a positive number mean?

Please select an answer first.

Correct answer: A

The correct answer is The nonnegative number whose square equals the given number..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: For example, the principal square root of 49 is 7, even though both 7 and -7 have a square of 49.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

For example, the principal square root of 49 is 7, even though both 7 and -7 have a square of 49.

10. QuestionMultiple choice

What is the exact value of 2⋅8\sqrt{2}\cdot\sqrt{8}?

Please select an answer first.

Correct answer: B

The correct answer is 44.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 2⋅8=16=4\sqrt{2}\cdot\sqrt{8}=\sqrt{16}=4.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

2⋅8=16=4\sqrt{2}\cdot\sqrt{8}=\sqrt{16}=4.

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