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Radicals and Real Numbers Practice Tests

These Grade 9 Math practice tests cover rational and irrational numbers, real-number classification, square roots, cube roots, principal roots, radical simplification, combining like radicals, exact and approximate values, rational denominators, Pythagorean applications, and real-number properties.

Skills covered

  • Classifying integers, rational numbers, irrational numbers, and real numbers
  • Evaluating and estimating square roots and cube roots
  • Simplifying, adding, subtracting, multiplying, and dividing radicals
  • Using exact radical forms in distance, diagonal, and right-triangle problems
  • Applying commutative, associative, distributive, identity, inverse, and closure properties

How to use these practice tests

Look for perfect-square or perfect-cube factors, keep the principal square root nonnegative, compare radicals by nearby perfect powers, and preserve exact forms unless a decimal approximation is requested.

For more Grade 9 Math practice, return to the Grade 9 Math category.

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1. Test

Grade 9 Radicals and Real Numbers Practice Tests

Practice Grade 9 Math questions about rational and irrational numbers, real-number properties, square roots, cube roots, simplifying radicals, exact values, approximations, and applications.

Choose the best answer. Classify the number, preserve exact radical forms when requested, estimate only when appropriate, and check signs and units in applications.

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

What is a rational number?

Please select an answer first.

Correct answer: A

The correct answer is A number that can be written as a fraction of two integers with a nonzero denominator..

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: Rational numbers include integers, terminating decimals, repeating decimals, and fractions of integers.
  3. Check the value and sign: The correct choice is A number that can be written as a fraction of two integers with a nonzero denominator..

Rational numbers include integers, terminating decimals, repeating decimals, and fractions of integers.

2. QuestionMultiple choice

Which number is irrational?

Please select an answer first.

Correct answer: B

The correct answer is 2\sqrt{2}.

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: The square root of 2 cannot be written as a ratio of integers and its decimal does not terminate or repeat.
  3. Check the value and sign: The correct choice is 2\sqrt{2}.

The square root of 2 cannot be written as a ratio of integers and its decimal does not terminate or repeat.

3. QuestionMultiple choice

Which statement about integers is true?

Please select an answer first.

Correct answer: C

The correct answer is Every integer is also a rational number..

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: Any integer n can be written as n/1, so every integer is rational.
  3. Check the value and sign: The correct choice is Every integer is also a rational number..

Any integer n can be written as n/1, so every integer is rational.

4. QuestionMultiple choice

What are the real numbers?

Please select an answer first.

Correct answer: D

The correct answer is All rational and irrational numbers on the number line..

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: The real number system contains both rational and irrational numbers.
  3. Check the value and sign: The correct choice is All rational and irrational numbers on the number line..

The real number system contains both rational and irrational numbers.

5. QuestionMultiple choice

Classify 49\sqrt{49} as precisely as possible.

Please select an answer first.

Correct answer: A

The correct answer is An integer..

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: The principal square root of 49 is 7, and 7 is an integer and therefore also rational and real.
  3. Check the value and sign: The correct choice is An integer..

The principal square root of 49 is 7, and 7 is an integer and therefore also rational and real.

6. QuestionMultiple choice

Why is 50\sqrt{50} irrational?

Please select an answer first.

Correct answer: B

The correct answer is Because 50 is not a perfect square..

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: Since 50 is not a perfect square, its square root cannot be written as a ratio of integers and is irrational.
  3. Check the value and sign: The correct choice is Because 50 is not a perfect square..

Since 50 is not a perfect square, its square root cannot be written as a ratio of integers and is irrational.

7. QuestionMultiple choice

What type of number is 0.333... with the 3 repeating?

Please select an answer first.

Correct answer: C

The correct answer is A rational number..

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: A repeating decimal can be written as a fraction, so it is rational.
  3. Check the value and sign: The correct choice is A rational number..

A repeating decimal can be written as a fraction, so it is rational.

8. QuestionMultiple choice

What type of number is π\pi?

Please select an answer first.

Correct answer: D

The correct answer is An irrational real number..

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: Pi is real but its decimal expansion is nonterminating and nonrepeating, so it is irrational.
  3. Check the value and sign: The correct choice is An irrational real number..

Pi is real but its decimal expansion is nonterminating and nonrepeating, so it is irrational.

9. QuestionMultiple choice

Between which two consecutive integers does 10\sqrt{10} lie?

Please select an answer first.

Correct answer: A

The correct answer is Between 3 and 4..

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: Because 9 < 10 < 16, taking square roots gives 3 < sqrt(10) < 4.
  3. Check the value and sign: The correct choice is Between 3 and 4..

Because 9 < 10 < 16, taking square roots gives 3 < sqrt(10) < 4.

10. QuestionMultiple choice

How are 16\sqrt{16} and 4 related?

Please select an answer first.

Correct answer: B

The correct answer is They are equal..

  1. Classify or represent the number: Determine the number set, radical form, exact value, approximation, or property involved.
  2. Apply the radical or real-number rule: The principal square root of 16 is the positive number 4.
  3. Check the value and sign: The correct choice is They are equal..

The principal square root of 16 is the positive number 4.

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