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

Book overview

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Unit 12: Area, Surface Area, and Volume

Back to Reveal Math, Accelerated

Lesson 1
Lesson 12-1: Describe Cross Sections of Three-Dimensional Figures
Learn Practice
Lesson 2
Lesson 12-2: Understand and Use Square Roots
Learn Practice
Lesson 3
Lesson 12-3: Solve Problems Involving Area and Surface Area
Learn Practice
Lesson 4
Lesson 12-4: Solve Problems Involving Circumference of Circles
Learn Practice
Lesson 5
Lesson 12-5: Solve Problems Involving Area of Circles
Learn Practice
Lesson 6Current
Lesson 12-6: Understand and Use Cube Roots
Learn Practice
Lesson 7
Lesson 12-7: Solve Problems Involving Volume of Cylinders and Cones
Learn Practice
Lesson 8
Lesson 12-8: Solve Problems Involving Volume of Spheres
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

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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

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Unit 12: Area, Surface Area, and Volume

Back to Reveal Math, Accelerated

Lesson 1
Lesson 12-1: Describe Cross Sections of Three-Dimensional Figures
Learn Practice
Lesson 2
Lesson 12-2: Understand and Use Square Roots
Learn Practice
Lesson 3
Lesson 12-3: Solve Problems Involving Area and Surface Area
Learn Practice
Lesson 4
Lesson 12-4: Solve Problems Involving Circumference of Circles
Learn Practice
Lesson 5
Lesson 12-5: Solve Problems Involving Area of Circles
Learn Practice
Lesson 6Current
Lesson 12-6: Understand and Use Cube Roots
Learn Practice
Lesson 7
Lesson 12-7: Solve Problems Involving Volume of Cylinders and Cones
Learn Practice
Lesson 8
Lesson 12-8: Solve Problems Involving Volume of Spheres
Learn Practice

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}}

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Unit 12: Area, Surface Area, and Volume

Back to Reveal Math, Accelerated

Lesson 1
Lesson 12-1: Describe Cross Sections of Three-Dimensional Figures
Learn Practice
Lesson 2
Lesson 12-2: Understand and Use Square Roots
Learn Practice
Lesson 3
Lesson 12-3: Solve Problems Involving Area and Surface Area
Learn Practice
Lesson 4
Lesson 12-4: Solve Problems Involving Circumference of Circles
Learn Practice
Lesson 5
Lesson 12-5: Solve Problems Involving Area of Circles
Learn Practice
Lesson 6Current
Lesson 12-6: Understand and Use Cube Roots
Learn Practice
Lesson 7
Lesson 12-7: Solve Problems Involving Volume of Cylinders and Cones
Learn Practice
Lesson 8
Lesson 12-8: Solve Problems Involving Volume of Spheres
Learn Practice

Lesson 12-6: Understand and Use Cube Roots

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Unit 12: Area, Surface Area, and Volume

Lesson 12-1: Describe Cross Sections of Three-Dimensional Figures

Lesson 12-2: Understand and Use Square Roots

Lesson 12-3: Solve Problems Involving Area and Surface Area

Lesson 12-4: Solve Problems Involving Circumference of Circles

Lesson 12-5: Solve Problems Involving Area of Circles

Lesson 12-6: Understand and Use Cube Roots

Lesson 12-7: Solve Problems Involving Volume of Cylinders and Cones

Lesson 12-8: Solve Problems Involving Volume of Spheres

Lesson overview

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Unit 12: Area, Surface Area, and Volume

Lesson 12-1: Describe Cross Sections of Three-Dimensional Figures

Lesson 12-2: Understand and Use Square Roots

Lesson 12-3: Solve Problems Involving Area and Surface Area

Lesson 12-4: Solve Problems Involving Circumference of Circles

Lesson 12-5: Solve Problems Involving Area of Circles

Lesson 12-6: Understand and Use Cube Roots

Lesson 12-7: Solve Problems Involving Volume of Cylinders and Cones

Lesson 12-8: Solve Problems Involving Volume of Spheres

Lesson 12-1: Describe Cross Sections of Three-Dimensional Figures

Lesson 12-2: Understand and Use Square Roots

Lesson 12-3: Solve Problems Involving Area and Surface Area

Lesson 12-4: Solve Problems Involving Circumference of Circles

Lesson 12-5: Solve Problems Involving Area of Circles

Lesson 12-6: Understand and Use Cube Roots

Lesson 12-7: Solve Problems Involving Volume of Cylinders and Cones

Lesson 12-8: Solve Problems Involving Volume of Spheres