Algebra 2 · Rational exponents and radicals

How to Solve an Equation with a Cube Root

Problem

\sqrt[3]{x - 1} = 2

Answer

x = 9

A cube root equation works like a square root equation, but you cube both sides. Unlike squaring, cubing never creates a false solution, because every real number has exactly one cube root.

Step-by-step solution

Cube both sides.
$\left(\sqrt[3]{x - 1}\right)^3 = 2^3$
Why: Cubing undoes a cube root.
Simplify both sides.
$x - 1 = 8$
Why: The left side becomes x - 1 and 2 cubed is 8.
Add 1 to both sides.
$x = 9$
Why: Adding undoes the subtraction.
Check the answer.
$\sqrt[3]{9 - 1} = \sqrt[3]{8} = 2$
Why: Substituting the answer back confirms it works.

Common mistakes

Squaring instead of cubing. Match the root you need to undo.
Writing plus or minus after cubing. A cube has exactly one real root.
Forgetting to add 1 back after cubing.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$\sqrt[3]{x} = 3$

Show answer

$x = 27$
$\sqrt[3]{x + 2} = 4$

Show answer

$x = 62$
$\sqrt[3]{2x - 1} = 3$

Show answer

$x = 14$
$\sqrt[3]{x - 5} = 2$

Show answer

$x = 13$
$\sqrt[3]{x + 1} = 5$

Show answer

$x = 124$