Cube Roots Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Cube Roots.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
The cube root of x, written \sqrt[3]{x}, is the number that when multiplied by itself three times equals x. Unlike square roots, cube roots are defined for negative numbers.
\sqrt[3]{27} asks: what number times itself times itself equals 27? Answer: 3, because 3 \times 3 \times 3 = 27. For negatives, \sqrt[3]{-8} = -2 because (-2) \times (-2) \times (-2) = -8.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Cube roots undo cubing, just as square roots undo squaring—but cube roots work for negative numbers too because a negative times a negative times a negative is negative.
Common stuck point: Remembering that cube roots of negative numbers exist and are negative, unlike square roots which are not real for negative inputs.
Sense of Study hint: Try guessing small whole numbers and cubing them: 1, 2, 3, 4, 5... to build a mental list of perfect cubes (1, 8, 27, 64, 125).
Worked Examples
Example 1
easySolution
- 1 A cube root asks for the number whose cube equals the expression inside the radical, so we want x^3 = -125.
- 2 Test a likely integer: (-5)^3 = (-5)(-5)(-5) = 25 \times (-5) = -125.
- 3 Therefore \sqrt[3]{-125} = -5.
Answer
Example 2
mediumExample 3
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyExample 2
easyRelated Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.