Lesson 2: Finding Cube Roots

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

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:

Examples

Property

To evaluate an algebraic expression involving a cube root for a given value of a variable, substitute the value for the variable and simplify the expression using the order of operations.

Examples

• Evaluate $5\sqrt[3]{x}$ when $x = 8$:

• Evaluate $\sqrt[3]{x + 19}$ when $x = 8$:

• Evaluate $10 - 4\sqrt[3]{x}$ when $x = -64$:

Explanation

This skill combines substitution with evaluating cube roots. First, replace the variable in the expression with its given numerical value. Next, follow the order of operations to simplify the expression. Remember to calculate the value of the cube root before performing any multiplication, division, addition, or subtraction outside the radical.