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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

Which number is greatest?

Please select an answer first.

Correct answer: A

The correct answer is 196\frac{19}{6}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 196≈3.167\frac{19}{6}\approx3.167, while 10≈3.162\sqrt{10}\approx3.162 and 8≈2.828\sqrt{8}\approx2.828.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

196≈3.167\frac{19}{6}\approx3.167, while 10≈3.162\sqrt{10}\approx3.162 and 8≈2.828\sqrt{8}\approx2.828.

2. QuestionMultiple choice

Which statement is true about squaring an irrational number?

Please select an answer first.

Correct answer: B

The correct answer is The square of an irrational number can be rational, such as (7)2=7(\sqrt{7})^2=7..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: Squaring can remove a radical and produce an integer or another rational number.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

Squaring can remove a radical and produce an integer or another rational number.

3. QuestionMultiple choice

What is the value of 273\sqrt[3]{27}?

Please select an answer first.

Correct answer: C

The correct answer is 33.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 3⋅3⋅3=273\cdot3\cdot3=27, so the cube root of 27 is 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.

3⋅3⋅3=273\cdot3\cdot3=27, so the cube root of 27 is 3.

4. QuestionMultiple choice

Between which two consecutive integers is 503\sqrt[3]{50}?

Please select an answer first.

Correct answer: D

The correct answer is Between 3 and 4.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: Because 33=273^3=27 and 43=644^3=64, 503\sqrt[3]{50} lies between 3 and 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.

Because 33=273^3=27 and 43=644^3=64, 503\sqrt[3]{50} lies between 3 and 4.

5. QuestionMultiple choice

Which type of number is 0.333…0.333\ldots?

Please select an answer first.

Correct answer: A

The correct answer is Rational.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 0.333…=130.333\ldots=\frac{1}{3}, so the repeating decimal is rational.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

0.333…=130.333\ldots=\frac{1}{3}, so the repeating decimal is rational.

6. QuestionMultiple choice

Which expression is an exact representation of the diagonal of a square with side length 1?

Please select an answer first.

Correct answer: B

The correct answer is 2\sqrt{2}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: The Pythagorean theorem gives d2=12+12=2d^2=1^2+1^2=2, so the exact diagonal is 2\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.

The Pythagorean theorem gives d2=12+12=2d^2=1^2+1^2=2, so the exact diagonal is 2\sqrt{2}.

7. QuestionMultiple choice

Which number is irrational?

Please select an answer first.

Correct answer: C

The correct answer is 13\sqrt{13}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 1313 is not a perfect square, so 13\sqrt{13} is irrational.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

1313 is not a perfect square, so 13\sqrt{13} is irrational.

8. QuestionMultiple choice

Which comparison is true?

Please select an answer first.

Correct answer: D

The correct answer is −3<−1.7-\sqrt{3}<-1.7.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 3≈1.732\sqrt{3}\approx1.732, so its opposite is about -1.732, which is less than -1.7.
  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≈1.732\sqrt{3}\approx1.732, so its opposite is about -1.732, which is less than -1.7.

9. QuestionMultiple choice

What is the distance between 2\sqrt{2} and 1 on the number line?

Please select an answer first.

Correct answer: A

The correct answer is 2−1\sqrt{2}-1.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: Because 2>1\sqrt{2}>1, the distance is ∣2−1∣=2−1|\sqrt{2}-1|=\sqrt{2}-1.
  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 2>1\sqrt{2}>1, the distance is ∣2−1∣=2−1|\sqrt{2}-1|=\sqrt{2}-1.

10. QuestionMultiple choice

Which statement about the estimate 2≈1.414\sqrt{2}\approx1.414 is true?

Please select an answer first.

Correct answer: B

The correct answer is The approximation is close to the exact value, but it is rounded and not equal to 2\sqrt{2}..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 1.4142=1.9993961.414^2=1.999396, which is close to 2, but a finite decimal cannot equal the irrational number 2\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.

1.4142=1.9993961.414^2=1.999396, which is close to 2, but a finite decimal cannot equal the irrational number 2\sqrt{2}.

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