Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 4: Real Numbers and the Pythagorean Theorem

Lesson 2: Finding Cube Roots

In this Grade 7 lesson from Big Ideas Math, Course 2 Accelerated, students learn how to find cube roots of perfect cubes using the cube root symbol and prime factorization. The lesson covers evaluating expressions involving cube roots and solving equations, building on students' prior knowledge of square roots to understand how cubing and taking a cube root are inverse operations. Students also explore why negative numbers can have cube roots, unlike square roots, aligning with the 8.EE.2 standard.

Section 1

Cube Root

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. Simplifying radicals occurs at the same level as powers in the order of operations.

Examples

To simplify $3\sqrt[3]{-8}$ , we find the cube root of $-8$ which is $-2$ , and then multiply by $3$ . So, $3\sqrt[3]{-8} = 3(-2) = -6$ .
To evaluate $2 - \sqrt[3]{-125}$ , we first find that the cube root of $-125$ is $-5$ . The expression becomes $2 - (-5) = 7$ .
To simplify $\frac{10 + \sqrt[3]{-27}}{7}$ , we calculate $\sqrt[3]{-27} = -3$ . The expression becomes $\frac{10 + (-3)}{7} = \frac{7}{7} = 1$ .

Explanation

A cube root is the inverse operation of cubing a number. Think of it as asking: 'What number, when multiplied by itself three times, gives me this value?' Unlike square roots, you can take the cube root of negative numbers.

Section 2

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

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Cube Root

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. Simplifying radicals occurs at the same level as powers in the order of operations.

Examples

To simplify $3\sqrt[3]{-8}$ , we find the cube root of $-8$ which is $-2$ , and then multiply by $3$ . So, $3\sqrt[3]{-8} = 3(-2) = -6$ .
To evaluate $2 - \sqrt[3]{-125}$ , we first find that the cube root of $-125$ is $-5$ . The expression becomes $2 - (-5) = 7$ .
To simplify $\frac{10 + \sqrt[3]{-27}}{7}$ , we calculate $\sqrt[3]{-27} = -3$ . The expression becomes $\frac{10 + (-3)}{7} = \frac{7}{7} = 1$ .

Explanation

Section 2

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

Overview

Lesson overview

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

Overview

Section 1

Cube Root

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. Simplifying radicals occurs at the same level as powers in the order of operations.

Examples

To simplify $3\sqrt[3]{-8}$ , we find the cube root of $-8$ which is $-2$ , and then multiply by $3$ . So, $3\sqrt[3]{-8} = 3(-2) = -6$ .
To evaluate $2 - \sqrt[3]{-125}$ , we first find that the cube root of $-125$ is $-5$ . The expression becomes $2 - (-5) = 7$ .
To simplify $\frac{10 + \sqrt[3]{-27}}{7}$ , we calculate $\sqrt[3]{-27} = -3$ . The expression becomes $\frac{10 + (-3)}{7} = \frac{7}{7} = 1$ .

Explanation

Section 2

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}