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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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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

Which is an exact form for the diagonal of a square with side length 1?

Please select an answer first.

Correct answer: A

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 Pythagorean theorem gives d squared = 1 squared + 1 squared = 2, so d = sqrt(2) exactly.
  3. Check the value and sign: The correct choice is 2\sqrt{2}.

The Pythagorean theorem gives d squared = 1 squared + 1 squared = 2, so d = sqrt(2) exactly.

2. QuestionMultiple choice

What is the distance between (0, 0) and (3, 4)?

Please select an answer first.

Correct answer: B

The correct answer is 5.

  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: Use the distance formula or a 3-4-5 right triangle: sqrt(3 squared + 4 squared) = 5.
  3. Check the value and sign: The correct choice is 5.

Use the distance formula or a 3-4-5 right triangle: sqrt(3 squared + 4 squared) = 5.

3. QuestionMultiple choice

A right triangle has hypotenuse 13 and one leg 5. What is the other leg?

Please select an answer first.

Correct answer: C

The correct answer is 12.

  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: Use a squared + 5 squared = 13 squared, so a squared = 144 and a = 12.
  3. Check the value and sign: The correct choice is 12.

Use a squared + 5 squared = 13 squared, so a squared = 144 and a = 12.

4. QuestionMultiple choice

What is the diagonal of a square with side length 6?

Please select an answer first.

Correct answer: D

The correct answer is 626\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 diagonal is side times sqrt(2), so it is 6sqrt(2).
  3. Check the value and sign: The correct choice is 626\sqrt{2}.

The diagonal is side times sqrt(2), so it is 6sqrt(2).

5. QuestionMultiple choice

Rewrite 12\frac{1}{\sqrt{2}} with a rational denominator.

Please select an answer first.

Correct answer: A

The correct answer is 22\frac{\sqrt{2}}{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: Multiply numerator and denominator by sqrt(2) to get sqrt(2)/2.
  3. Check the value and sign: The correct choice is 22\frac{\sqrt{2}}{2}.

Multiply numerator and denominator by sqrt(2) to get sqrt(2)/2.

6. QuestionMultiple choice

Simplify 27+12\sqrt{27}+\sqrt{12}

Please select an answer first.

Correct answer: B

The correct answer is 535\sqrt{3}.

  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: Rewrite as 3sqrt(3) + 2sqrt(3), then add like radicals.
  3. Check the value and sign: The correct choice is 535\sqrt{3}.

Rewrite as 3sqrt(3) + 2sqrt(3), then add like radicals.

7. QuestionMultiple choice

Simplify 26⋅322\sqrt{6}\cdot3\sqrt{2}

Please select an answer first.

Correct answer: C

The correct answer is 12312\sqrt{3}.

  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: Multiply coefficients and radicals: 6sqrt(12) = 6(2sqrt(3)) = 12sqrt(3).
  3. Check the value and sign: The correct choice is 12312\sqrt{3}.

Multiply coefficients and radicals: 6sqrt(12) = 6(2sqrt(3)) = 12sqrt(3).

8. QuestionMultiple choice

Simplify (5)2(\sqrt{5})^2

Please select an answer first.

Correct answer: D

The correct answer is 5.

  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: Squaring a principal square root returns the original nonnegative radicand.
  3. Check the value and sign: The correct choice is 5.

Squaring a principal square root returns the original nonnegative radicand.

9. QuestionMultiple choice

Which is greater, 7\sqrt{7} or 2.5?

Please select an answer first.

Correct answer: A

The correct answer is 7\sqrt{7}.

  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 sqrt(7) is approximately 2.646, it is greater than 2.5.
  3. Check the value and sign: The correct choice is 7\sqrt{7}.

Since sqrt(7) is approximately 2.646, it is greater than 2.5.

10. QuestionMultiple choice

Approximate 50\sqrt{50} to the nearest tenth.

Please select an answer first.

Correct answer: B

The correct answer is 7.1.

  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 sqrt(50) is approximately 7.071, it rounds to 7.1.
  3. Check the value and sign: The correct choice is 7.1.

Since sqrt(50) is approximately 7.071, it rounds to 7.1.

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