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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 type of number is the sum of a rational number and an irrational number?

Please select an answer first.

Correct answer: A

The correct answer is It is irrational..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: If a rational plus an irrational were rational, subtracting the rational would make the irrational rational, which is impossible.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

If a rational plus an irrational were rational, subtracting the rational would make the irrational rational, which is impossible.

2. QuestionMultiple choice

What is the value of 3+(−3)\sqrt{3}+(-\sqrt{3})?

Please select an answer first.

Correct answer: B

The correct answer is 00.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: The opposite values cancel: 3−3=0\sqrt{3}-\sqrt{3}=0.
  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 opposite values cancel: 3−3=0\sqrt{3}-\sqrt{3}=0.

3. QuestionMultiple choice

What is the value of (3)2(\sqrt{3})^2?

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: Squaring a principal square root returns the original nonnegative number: (3)2=3(\sqrt{3})^2=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.

Squaring a principal square root returns the original nonnegative number: (3)2=3(\sqrt{3})^2=3.

4. QuestionMultiple choice

If rr is a nonzero rational number and xx is irrational, what type of number is rxrx?

Please select an answer first.

Correct answer: D

The correct answer is Irrational.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: A nonzero rational multiplier cannot turn an irrational number into a rational number; otherwise dividing by rr would make xx 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.

A nonzero rational multiplier cannot turn an irrational number into a rational number; otherwise dividing by rr would make xx rational.

5. QuestionMultiple choice

What is the simplified form of 2+8\sqrt{2}+\sqrt{8}?

Please select an answer first.

Correct answer: A

The correct answer is 323\sqrt{2}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 8=22\sqrt{8}=2\sqrt{2}, so 2+22=32\sqrt{2}+2\sqrt{2}=3\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.

8=22\sqrt{8}=2\sqrt{2}, so 2+22=32\sqrt{2}+2\sqrt{2}=3\sqrt{2}.

6. QuestionMultiple choice

What is the simplified form of 18−8\sqrt{18}-\sqrt{8}?

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: 18=32\sqrt{18}=3\sqrt{2} and 8=22\sqrt{8}=2\sqrt{2}, so the difference 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.

18=32\sqrt{18}=3\sqrt{2} and 8=22\sqrt{8}=2\sqrt{2}, so the difference is 2\sqrt{2}.

7. QuestionMultiple choice

What is the value of (5)2(\sqrt{5})^2?

Please select an answer first.

Correct answer: C

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: The square and principal square-root operations undo each other for a nonnegative 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.

The square and principal square-root operations undo each other for a nonnegative number.

8. QuestionMultiple choice

What is the value of 2⋅18\sqrt{2}\cdot\sqrt{18}?

Please select an answer first.

Correct answer: D

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: 2⋅18=36=6\sqrt{2}\cdot\sqrt{18}=\sqrt{36}=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.

2⋅18=36=6\sqrt{2}\cdot\sqrt{18}=\sqrt{36}=6.

9. QuestionMultiple choice

Which property is true for the real numbers?

Please select an answer first.

Correct answer: A

The correct answer is The sum or product of two real numbers is always a real number..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: The real numbers are closed under addition and multiplication, so these operations produce real results.
  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 real numbers are closed under addition and multiplication, so these operations produce real results.

10. QuestionMultiple choice

Why is an exact radical form often useful?

Please select an answer first.

Correct answer: B

The correct answer is It preserves the precise value instead of replacing it with a rounded decimal..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 2\sqrt{2} is exact, while 1.414 is only an approximation.
  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\sqrt{2} is exact, while 1.414 is only an approximation.

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