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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

What are the solutions of x2=49x^2=49?

Please select an answer first.

Correct answer: A

The correct answer is x=±7x=\pm7.

  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 7 and -7 have a square of 49, so the solutions are x=7x=7 and x=−7x=-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 7 and -7 have a square of 49, so the solutions are x=7x=7 and x=−7x=-7.

2. QuestionMultiple choice

What is the solution of x2=25x^2=25 when xx represents a length?

Please select an answer first.

Correct answer: B

The correct answer is x=5x=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: A length is nonnegative, so use the principal square root: x=25=5x=\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.

A length is nonnegative, so use the principal square root: x=25=5x=\sqrt{25}=5.

3. QuestionMultiple choice

What is the solution of x3=64x^3=64?

Please select an answer first.

Correct answer: C

The correct answer is x=4x=4.

  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: 43=644^3=64, and a cube equation has one real cube-root solution here.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

43=644^3=64, and a cube equation has one real cube-root solution here.

4. QuestionMultiple choice

What is the solution of x3=−125x^3=-125?

Please select an answer first.

Correct answer: D

The correct answer is x=−5x=-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: (−5)3=−125(-5)^3=-125, so the real cube-root solution is -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.

(−5)3=−125(-5)^3=-125, so the real cube-root solution is -5.

5. QuestionMultiple choice

Which equation has solutions x=±10x=\pm\sqrt{10}?

Please select an answer first.

Correct answer: A

The correct answer is x2=10x^2=10.

  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: Squaring both solutions gives x2=10x^2=10, so the equation has the two roots ±10\pm\sqrt{10}.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

Squaring both solutions gives x2=10x^2=10, so the equation has the two roots ±10\pm\sqrt{10}.

6. QuestionMultiple choice

If the area of a square is 81 square units81\text{ square units}, what is its side length?

Please select an answer first.

Correct answer: B

The correct answer is 99 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: The side length is the principal square root of the area: 81=9\sqrt{81}=9.
  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 side length is the principal square root of the area: 81=9\sqrt{81}=9.

7. QuestionMultiple choice

A cube has volume 343 cubic units343\text{ cubic units}. What is its edge length?

Please select an answer first.

Correct answer: C

The correct answer is 77 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: The edge length is the cube root of the volume: 3433=7\sqrt[3]{343}=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.

The edge length is the cube root of the volume: 3433=7\sqrt[3]{343}=7.

8. QuestionMultiple choice

A right triangle has legs 6 and 8. What is the exact hypotenuse?

Please select an answer first.

Correct answer: D

The correct answer is 1010.

  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: c=62+82=36+64=100=10c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

c=62+82=36+64=100=10c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10.

9. QuestionMultiple choice

A square has diagonal 10210\sqrt{2} units. What is its side length?

Please select an answer first.

Correct answer: A

The correct answer is 1010 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: For a square, d=s2d=s\sqrt{2}, so s=1022=10s=\frac{10\sqrt{2}}{\sqrt{2}}=10.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

For a square, d=s2d=s\sqrt{2}, so s=1022=10s=\frac{10\sqrt{2}}{\sqrt{2}}=10.

10. QuestionMultiple choice

Why is x=±ax=\pm\sqrt{a} used for x2=ax^2=a when a>0a>0?

Please select an answer first.

Correct answer: B

The correct answer is Both a positive and a negative number have the same positive square..

  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: For positive aa, both a\sqrt{a} and −a-\sqrt{a} square to aa.
  3. Check the result: Square or cube the answer when possible, verify the sign, and keep exact radical form unless an approximation is requested.

For positive aa, both a\sqrt{a} and −a-\sqrt{a} square to aa.

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