Lesson 2: The Cube Root Function

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. 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.

Property

The cube root function $f(x) = \sqrt[3]{x}$ is always increasing (monotonically increasing) on its entire domain. This means that for any two values $x_1 < x_2$, we have $f(x_1) < f(x_2)$, or equivalently: if $a < b$, then $\sqrt[3]{a} < \sqrt[3]{b}$.

Examples