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Square Roots and Cube Roots Practice Tests

These Grade 8 Math practice tests cover square roots, cube roots, perfect squares, perfect cubes, radical simplification, estimates, equations, comparisons, and geometry applications.

Skills covered

  • Evaluating principal square roots and cube roots using perfect squares and perfect cubes
  • Locating and estimating radicals with nearby powers and number-line reasoning
  • Simplifying square-root and cube-root expressions while preserving exact values
  • Solving equations such as \(x^2=a\) and \(x^3=a\)
  • Applying roots to area, volume, distance, diagonal, and right-triangle problems

How to use these practice tests

Find nearby perfect squares or cubes first. Then simplify or estimate the radical, check the sign, and square or cube the result to verify it.

For more Grade 8 Math practice, return to the Grade 8 Math category.

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Grade 8 Square Roots and Cube Roots Practice Tests

Practice Grade 8 Math questions about square roots, cube roots, perfect squares, perfect cubes, radical simplification, estimates, equations, and geometry applications.

Choose the best answer for each question. Keep exact radical form 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 side length 3\sqrt{3} units. What is its area?

Please select an answer first.

Correct answer: A

The correct answer is 3 square units3\text{ square units}.

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: A=s2=(3)2=3A=s^2=(\sqrt{3})^2=3, so the area is 3 square units.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

A=s2=(3)2=3A=s^2=(\sqrt{3})^2=3, so the area is 3 square units.

2. QuestionMultiple choice

A cube has edge length 53\sqrt[3]{5} units. What is its volume?

Please select an answer first.

Correct answer: B

The correct answer is 5 cubic units5\text{ cubic units}.

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: V=s3=(53)3=5V=s^3=(\sqrt[3]{5})^3=5.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

V=s3=(53)3=5V=s^3=(\sqrt[3]{5})^3=5.

3. QuestionMultiple choice

Which expression gives the distance between the points 22 and 7\sqrt{7} on the number line?

Please select an answer first.

Correct answer: C

The correct answer is 7−2\sqrt{7}-2.

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: Since 7≈2.646>2\sqrt{7}\approx2.646>2, the distance is ∣7−2∣=7−2|\sqrt{7}-2|=\sqrt{7}-2.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

Since 7≈2.646>2\sqrt{7}\approx2.646>2, the distance is ∣7−2∣=7−2|\sqrt{7}-2|=\sqrt{7}-2.

4. QuestionMultiple choice

Which equation represents a square with area 1818 square units?

Please select an answer first.

Correct answer: D

The correct answer is s2=18s^2=18.

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: The area of a square is the side length squared, so s2=18s^2=18 and the positive side is 18\sqrt{18}.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

The area of a square is the side length squared, so s2=18s^2=18 and the positive side is 18\sqrt{18}.

5. QuestionMultiple choice

Which is the simplified form of 2122\sqrt{12}?

Please select an answer first.

Correct answer: A

The correct answer is 434\sqrt{3}.

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: 212=2(23)=432\sqrt{12}=2(2\sqrt{3})=4\sqrt{3}.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

212=2(23)=432\sqrt{12}=2(2\sqrt{3})=4\sqrt{3}.

6. QuestionMultiple choice

Which is the simplified form of 3203\sqrt{20}?

Please select an answer first.

Correct answer: B

The correct answer is 656\sqrt{5}.

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: 320=3(25)=653\sqrt{20}=3(2\sqrt{5})=6\sqrt{5}.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

320=3(25)=653\sqrt{20}=3(2\sqrt{5})=6\sqrt{5}.

7. QuestionMultiple choice

Which product is equal to 6⋅24\sqrt{6}\cdot\sqrt{24}?

Please select an answer first.

Correct answer: C

The correct answer is 1212.

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: 6⋅24=144=12\sqrt{6}\cdot\sqrt{24}=\sqrt{144}=12.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

6⋅24=144=12\sqrt{6}\cdot\sqrt{24}=\sqrt{144}=12.

8. QuestionMultiple choice

Which quotient is equal to 502\frac{\sqrt{50}}{\sqrt{2}}?

Please select an answer first.

Correct answer: D

The correct answer is 55.

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: 502=502=25=5\frac{\sqrt{50}}{\sqrt{2}}=\sqrt{\frac{50}{2}}=\sqrt{25}=5.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

502=502=25=5\frac{\sqrt{50}}{\sqrt{2}}=\sqrt{\frac{50}{2}}=\sqrt{25}=5.

9. QuestionMultiple choice

Which statement best explains why 12\sqrt{12} and 232\sqrt{3} are equal?

Please select an answer first.

Correct answer: A

The correct answer is Both expressions square to 12 and represent the same nonnegative principal root..

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: 12=4⋅3=23\sqrt{12}=\sqrt{4\cdot3}=2\sqrt{3}, so the two exact forms are equivalent.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

12=4⋅3=23\sqrt{12}=\sqrt{4\cdot3}=2\sqrt{3}, so the two exact forms are equivalent.

10. QuestionMultiple choice

What is the best strategy for comparing 48\sqrt{48} and 77?

Please select an answer first.

Correct answer: B

The correct answer is Compare their squares: 48 and 49, so 48<7\sqrt{48}<7..

  1. Identify the root or radical: Determine whether the problem asks for a square root, cube root, perfect power, estimate, simplification, comparison, or equation solution.
  2. Use nearby perfect powers: Both values are nonnegative, and because 48<49=7248<49=7^2, it follows that 48<7\sqrt{48}<7.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

Both values are nonnegative, and because 48<49=7248<49=7^2, it follows that 48<7\sqrt{48}<7.

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