--- title: "Cube Root | Intermediate Algebra" description: "Cube Root in Intermediate Algebra is the number that, when multiplied by itself three times, gives the original value, and it shows up in radical and equation solving." canonical: "https://fiveable.me/intermediate-algebra/key-terms/cube-root" type: "key-term" subject: "Intermediate Algebra" unit: "Unit 8" --- # Cube Root | Intermediate Algebra ## Definition A cube root is the number that, when multiplied by itself three times, equals the original number. In Intermediate Algebra, you use cube roots to simplify radicals, work with volume, and solve equations with third powers. ## What It Is A cube root in Intermediate Algebra is the inverse of cubing a number. If a number is cubed to make a value, the cube root takes you back to the original number. So 8 is the cube root of 512 because 8 b7 8 b7 8 = 512, and b2b2? Actually, the shorthand is 35128. The notation is root index 3, written as . Unlike square roots, cube roots can be negative because a negative number times itself three times stays negative. That means -8 = -2, since (-2)(-2)(-2) = -8. In Intermediate Algebra, you usually meet cube roots when simplifying radicals, solving radical equations, or working with formulas that come from volume. For a cube, volume is s^3, so finding the side length from a volume means taking a cube root. If a cube has volume 64 cubic units, each side is 364 = 4. A useful detail is that cube roots do not have the same restriction as even roots. The cube root of a negative real number is still real, so 3-27 = -3. That difference matters when you compare cube roots to square roots, because square roots of negative numbers are not real in this course. You will also simplify cube root expressions by pulling out perfect cubes. For example, 354 can be rewritten as 327 b7 2 = 332. This is the same move you use with square roots, just with perfect cubes instead of perfect squares. The index tells you what kind of power to look for. ## Why It Matters Cube roots show up anywhere Intermediate Algebra asks you to reverse a third power or simplify expressions with radicals. That includes solving equations like x^3 = 125, finding missing dimensions from volume formulas, and rewriting expressions so they are easier to combine or compare. They also connect to function work. When you graph or evaluate radical functions that use a cube root, you need to know that the graph can extend into negative x-values and negative outputs, unlike many square root graphs. That changes the domain, the shape of the graph, and how you read intercepts. This term also builds your exponent sense. If you can move smoothly between roots and powers, you are better at simplifying expressions, checking answers, and spotting when a radical equation has been solved correctly. Cube roots are a good checkpoint for whether you really understand inverse operations, not just memorized a rule. ## Connections ### Radical A cube root is a kind of radical, which means it is written with a root symbol instead of an exponent. In Intermediate Algebra, radicals show up in simplification, equations, and function graphs. Once you know the index on the radical, you know what power you are undoing. ### Exponent Cube roots and exponents are inverse operations. If x^3 gives you a number, the cube root takes you back to x. This connection matters when you solve equations by undoing a power, especially if the original problem started with a cube or a volume formula. ### Simplify Simplifying cube roots means breaking the radicand into a perfect cube times whatever is left. For example, 54 becomes 27 b7 2, so 354 simplifies to 332. This is the same style of factoring you use with other radicals, just with cube patterns. ### [Function](/intermediate-algebra/key-terms/function) Cube roots appear in radical functions like f(x) = 3 x. These graphs behave differently from square root graphs because they can take negative inputs and produce negative outputs. That makes cube root functions useful for graphing practice and for comparing how different radicals behave. ## On the AP Exam On quizzes and problem sets, you might be asked to evaluate a cube root, simplify one, or solve an equation such as x^3 = 216 by taking a cube root on both sides. You may also see a word problem about volume where you need the side length of a cube from its volume. Another common task is spotting whether a radical expression should stay as a cube root or be rewritten using a perfect cube factor. If the problem includes a negative radicand, cube roots are one of the few radical types where a real answer can still exist. That is a place where careful sign work matters. ## Cube Root vs Square Root Cube roots and square roots both undo powers, but they do not behave the same way. A square root asks what number times itself gives the radicand, while a cube root asks what number times itself three times gives the radicand. In Intermediate Algebra, cube roots can be negative and are used with third powers, while square roots stay nonnegative in the real-number system. ## Key Takeaways - A cube root is the number that, when multiplied by itself three times, gives the original number. - The notation 3 means cube root, and the 3 tells you the index. - Cube roots are the inverse of cubing, so they undo expressions like x^3. - Negative numbers can have real cube roots, which is different from square roots in this course. - You simplify cube roots by factoring out perfect cubes, just like you simplify other radicals by factoring out perfect powers. ## FAQs ### What is cube root in Intermediate Algebra? A cube root is the value that gives you the original number when multiplied by itself three times. In Intermediate Algebra, you use cube roots to reverse third powers, simplify radical expressions, and solve volume or radical-equation problems. ### How do you simplify a cube root? Look for a perfect cube inside the radicand, factor it out, and take its cube root. For example, 354 becomes 327 b7 2 = 332. The common mistake is pulling out numbers that are not perfect cubes. ### Is the cube root of a negative number real? Yes, in Intermediate Algebra it can be real. For example, 3-8 = -2 because (-2)(-2)(-2) = -8. That is different from even-index radicals, which do not have real answers for negative radicands. ### How do cube roots show up in equations? They show up when you solve a variable that was cubed. If x^3 = 64, you take the cube root of both sides to get x = 4. If the equation already has a cube root, first isolate it, then cube both sides to remove it. ## Related Study Guides - [8.2 Simplify Radical Expressions](/intermediate-algebra/unit-8/2-simplify-radical-expressions/study-guide/8EWeFEZXLEx1beLc) - [8.7 Use Radicals in Functions](/intermediate-algebra/unit-8/7-radicals-functions/study-guide/GIPlJ5KFp4p9y21i) - [8.6 Solve Radical Equations](/intermediate-algebra/unit-8/6-solve-radical-equations/study-guide/QpaDLas81vJbJIfy) - [8.5 Divide Radical Expressions](/intermediate-algebra/unit-8/5-divide-radical-expressions/study-guide/WNQFtlotcEzgNFDK) - [8.1 Simplify Expressions with Roots](/intermediate-algebra/unit-8/1-simplify-expressions-roots/study-guide/ar2bvSdPtyGp6N6m) - [8.4 Add, Subtract, and Multiply Radical Expressions](/intermediate-algebra/unit-8/4-add-subtract-multiply-radical-expressions/study-guide/bkuZSI4GIYmCEaLH) ## About This Document Canonical Fiveable pages are available as Markdown at the same path plus `.md`. - [llms.txt](https://fiveable.me/llms.txt): index of Fiveable's sections and URL patterns - [llms-full.txt](https://fiveable.me/llms-full.txt): complete subject and unit listing - [MCP server](https://fiveable.me/mcp): call Fiveable as tools instead of fetching pages (`https://fiveable.me/api/mcp`) - [MCP server for AP teachers](https://fiveable.me/mcp/teachers): a teacher's classes, assignments and AP-rubric grading (`https://fiveable.me/api/mcp/teacher`) ## Structured Data ```json {"@context":"https://schema.org","@graph":[{"@type":"LearningResource","@id":"https://fiveable.me/intermediate-algebra/key-terms/cube-root#resource","name":"Cube Root | Intermediate Algebra","url":"https://fiveable.me/intermediate-algebra/key-terms/cube-root","learningResourceType":"Concept explainer","educationalLevel":"AP® / High School","about":{"@id":"https://fiveable.me/intermediate-algebra/key-terms/cube-root#term"},"audience":{"@type":"EducationalAudience","educationalRole":"student"},"dateModified":"2026-07-03T02:21:59.351Z","isPartOf":{"@type":"Collection","name":"Intermediate Algebra Key Terms","url":"https://fiveable.me/intermediate-algebra/key-terms"},"publisher":{"@type":"Organization","name":"Fiveable","url":"https://fiveable.me"}},{"@type":"DefinedTerm","@id":"https://fiveable.me/intermediate-algebra/key-terms/cube-root#term","name":"Cube Root","description":"A cube root is the number that, when multiplied by itself three times, equals the original number. 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