Learn on PengiReveal Math, AcceleratedUnit 12: Area, Surface Area, and Volume

Lesson 12-6: Understand and Use Cube Roots

In this Grade 7 lesson from Reveal Math, Accelerated (Unit 12), students learn to understand and apply cube roots as the inverse operation of cubing a number, using the relationship s = ∛V to find the side length of a cube from its volume. Students identify perfect cubes, estimate cube roots of non-perfect cubes between consecutive whole numbers, and simplify cube roots of fractions. The lesson also connects cube roots to real-world contexts, including applying Kepler's Third Law to calculate planetary distances using the equation T² = r³.

Section 1

Cube Root Definition and Notation

Property

$b$ is the cube root of $a$ if $b$ cubed equals $a$ . In symbols, we write

b = \sqrt[3]{a} \quad \text{if} \quad b^3 = a

Unlike square roots, which are not real for negative numbers, every real number has a real cube root.

Examples

Section 2

Cube Roots of Perfect Cubes

Property

A perfect cube is a number that can be written as the cube (third power) of an integer. If a number $x$ is a perfect cube such that $x = a^3$ , then the cube root of $x$ is $a$ :

\sqrt[3]{x} = a

Examples

Section 3

Finding Cube Roots of Negative Numbers

Property

For any positive number $a$ , the cube root of $-a$ is the negative of the cube root of $a$ .

\sqrt[3]{-a} = -\sqrt[3]{a}

Examples

Section 4

Finding Cube Roots of Rational Numbers

Property

To find the cube root of a perfect cube integer, you can use its prime factorization. For a fraction, the cube root of the quotient is the quotient of the cube roots:

\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Cube Root Definition and Notation

Property

$b$ is the cube root of $a$ if $b$ cubed equals $a$ . In symbols, we write

b = \sqrt[3]{a} \quad \text{if} \quad b^3 = a

Unlike square roots, which are not real for negative numbers, every real number has a real cube root.

Examples

Section 2

Cube Roots of Perfect Cubes

Property

A perfect cube is a number that can be written as the cube (third power) of an integer. If a number $x$ is a perfect cube such that $x = a^3$ , then the cube root of $x$ is $a$ :

\sqrt[3]{x} = a

Examples

Section 3

Finding Cube Roots of Negative Numbers

Property

For any positive number $a$ , the cube root of $-a$ is the negative of the cube root of $a$ .

\sqrt[3]{-a} = -\sqrt[3]{a}

Examples

Section 4

Finding Cube Roots of Rational Numbers

Property

To find the cube root of a perfect cube integer, you can use its prime factorization. For a fraction, the cube root of the quotient is the quotient of the cube roots:

\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Cube Root Definition and Notation

Property

$b$ is the cube root of $a$ if $b$ cubed equals $a$ . In symbols, we write

b = \sqrt[3]{a} \quad \text{if} \quad b^3 = a

Unlike square roots, which are not real for negative numbers, every real number has a real cube root.

Examples

Section 2

Cube Roots of Perfect Cubes

Property

A perfect cube is a number that can be written as the cube (third power) of an integer. If a number $x$ is a perfect cube such that $x = a^3$ , then the cube root of $x$ is $a$ :

\sqrt[3]{x} = a

Examples

Section 3

Finding Cube Roots of Negative Numbers

Property

For any positive number $a$ , the cube root of $-a$ is the negative of the cube root of $a$ .

\sqrt[3]{-a} = -\sqrt[3]{a}

Examples

Section 4

Finding Cube Roots of Rational Numbers

Property

To find the cube root of a perfect cube integer, you can use its prime factorization. For a fraction, the cube root of the quotient is the quotient of the cube roots:

\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}