Section 7.2: Finding Cube Roots

Property

The exponent 3 is used frequently, so it has a special name.

Instead of reading $5^3$ as "5 raised to the third power," we say "5 cubed."

Property

The number $c$ is called a cube root of a number $b$ if $c^3 = b$. Every number has exactly one cube root. The cube root of a positive number is positive, and the cube root of a negative number is negative.

Examples

• The cube root of 27 is 3, written as $\sqrt[3]{27} = 3$, because $3^3 = 27$.
• The cube root of $-64$ is $-4$, written as $\sqrt[3]{-64} = -4$, because $(-4)^3 = -64$.
• A cube-shaped box has a volume of 1000 cubic inches. Its side length is the cube root of the volume, so $\sqrt[3]{1000} = 10$ inches.

Explanation

Finding a cube root is the reverse of cubing a number (raising it to the third power).

Property

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

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

Examples