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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 property says that the sum of two rational numbers is rational?

Please select an answer first.

Correct answer: A

The correct answer is Closure under addition..

  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 rational numbers are closed under addition because adding any two rational numbers gives another rational number.
  3. Check the value and sign: The correct choice is Closure under addition..

The rational numbers are closed under addition because adding any two rational numbers gives another rational number.

2. QuestionMultiple choice

Which property is shown by 3⋅5=5⋅33\cdot5=5\cdot3?

Please select an answer first.

Correct answer: B

The correct answer is The commutative property of multiplication..

  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: Changing the order of the factors does not change the product.
  3. Check the value and sign: The correct choice is The commutative property of multiplication..

Changing the order of the factors does not change the product.

3. QuestionMultiple choice

Which property is shown by (a+b)+c=a+(b+c)(a+b)+c=a+(b+c)?

Please select an answer first.

Correct answer: C

The correct answer is The associative property of addition..

  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 grouping changes but the order of the addends stays the same.
  3. Check the value and sign: The correct choice is The associative property of addition..

The grouping changes but the order of the addends stays the same.

4. QuestionMultiple choice

Which property is shown by a(b+c)=ab+aca(b+c)=ab+ac?

Please select an answer first.

Correct answer: D

The correct answer is The distributive property..

  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: Multiplication is distributed across the sum inside the parentheses.
  3. Check the value and sign: The correct choice is The distributive property..

Multiplication is distributed across the sum inside the parentheses.

5. QuestionMultiple choice

What is the additive inverse of -8?

Please select an answer first.

Correct answer: A

The correct answer is 8.

  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 number and its additive inverse sum to zero, so the additive inverse of -8 is 8.
  3. Check the value and sign: The correct choice is 8.

A number and its additive inverse sum to zero, so the additive inverse of -8 is 8.

6. QuestionMultiple choice

What is the multiplicative inverse of 5?

Please select an answer first.

Correct answer: B

The correct answer is 15\frac{1}{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: A nonzero number times its reciprocal equals 1.
  3. Check the value and sign: The correct choice is 15\frac{1}{5}.

A nonzero number times its reciprocal equals 1.

7. QuestionMultiple choice

Simplify 9x2\sqrt{9x^2} assuming x is nonnegative.

Please select an answer first.

Correct answer: C

The correct answer is 3x3x.

  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: sqrt(9x squared) = sqrt(9) times sqrt(x squared) = 3x when x is nonnegative.
  3. Check the value and sign: The correct choice is 3x3x.

sqrt(9x squared) = sqrt(9) times sqrt(x squared) = 3x when x is nonnegative.

8. QuestionMultiple choice

Solve x=7\sqrt{x}=7

Please select an answer first.

Correct answer: D

The correct answer is 49.

  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: Square both sides: x = 7 squared = 49.
  3. Check the value and sign: The correct choice is 49.

Square both sides: x = 7 squared = 49.

9. QuestionMultiple choice

Solve x+1=4\sqrt{x+1}=4

Please select an answer first.

Correct answer: A

The correct answer is 15.

  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: Square both sides: x + 1 = 16, so x = 15.
  3. Check the value and sign: The correct choice is 15.

Square both sides: x + 1 = 16, so x = 15.

10. QuestionMultiple choice

Which statement best summarizes the real number system?

Please select an answer first.

Correct answer: B

The correct answer is It includes rational and irrational numbers and can be represented on a 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: Real numbers include all rational and irrational values, including negative numbers and many radicals.
  3. Check the value and sign: The correct choice is It includes rational and irrational numbers and can be represented on a number line..

Real numbers include all rational and irrational values, including negative numbers and many radicals.

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