NumicornNumicornLearn, solve, progress

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.

Select a test

5 Tests
1. Test2. Test3. Test4. Test5. Test

5. Test

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

A square has area 2 square units2\text{ square units}. What is the exact side length?

Please select an answer first.

Correct answer: A

The correct answer is 2\sqrt{2} units.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: If the side length is ss, then s2=2s^2=2, so the positive side length is s=2s=\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.

If the side length is ss, then s2=2s^2=2, so the positive side length is s=2s=\sqrt{2}.

2. QuestionMultiple choice

What is the exact diagonal of a square with side length 11 unit?

Please select an answer first.

Correct answer: B

The correct answer is 2\sqrt{2} units.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: Using the Pythagorean theorem, d=12+12=2d=\sqrt{1^2+1^2}=\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.

Using the Pythagorean theorem, d=12+12=2d=\sqrt{1^2+1^2}=\sqrt{2}.

3. QuestionMultiple choice

A right triangle has legs of length 5 and 12. What is the hypotenuse?

Please select an answer first.

Correct answer: C

The correct answer is 1313.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: c=52+122=25+144=169=13c=\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

c=52+122=25+144=169=13c=\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13.

4. QuestionMultiple choice

Which statement about the circumference of a circle with radius 11 is true?

Please select an answer first.

Correct answer: D

The correct answer is Its exact circumference is 2π2\pi, an irrational number..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: The circumference formula is C=2πrC=2\pi r, so when r=1r=1, C=2πC=2\pi.
  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 circumference formula is C=2πrC=2\pi r, so when r=1r=1, C=2πC=2\pi.

5. QuestionMultiple choice

The average of two rational numbers is what type of number?

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: The average is their sum divided by 2, and rational numbers are closed under addition and division by a nonzero 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.

The average is their sum divided by 2, and rational numbers are closed under addition and division by a nonzero rational number.

6. QuestionMultiple choice

Which is the best reason to use 18\sqrt{18} instead of 4.244.24 when an exact answer is required?

Please select an answer first.

Correct answer: B

The correct answer is The radical form preserves the exact value..

  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} is exact, while 4.24 is a rounded 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.

18=32\sqrt{18}=3\sqrt{2} is exact, while 4.24 is a rounded approximation.

7. QuestionMultiple choice

Which expression is irrational?

Please select an answer first.

Correct answer: C

The correct answer is 7+1\sqrt{7}+1.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: Adding the rational number 1 to the irrational number 7\sqrt{7} produces an irrational 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.

Adding the rational number 1 to the irrational number 7\sqrt{7} produces an irrational number.

8. QuestionMultiple choice

What are the solutions of x2=7x^2=7?

Please select an answer first.

Correct answer: D

The correct answer is x=±7x=\pm\sqrt{7}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: Both 7\sqrt{7} and −7-\sqrt{7} have a square of 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.

Both 7\sqrt{7} and −7-\sqrt{7} have a square of 7.

9. QuestionMultiple choice

Which is the best estimate for 45\sqrt{45}?

Please select an answer first.

Correct answer: A

The correct answer is 6.76.7.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 6.72=44.896.7^2=44.89, which is close to 45, so 6.7 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.

6.72=44.896.7^2=44.89, which is close to 45, so 6.7 is a reasonable estimate.

10. QuestionMultiple choice

Which conclusion best describes the real-number system?

Please select an answer first.

Correct answer: B

The correct answer is It includes rational and irrational numbers and supports comparisons and operations on the number line..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: The real numbers form a broad system containing rational and irrational values, including positive, negative, and zero.
  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 form a broad system containing rational and irrational values, including positive, negative, and zero.

Previous testNext test