Cube Roots Formula

The Formula

\sqrt[3]{x} = y \iff y^3 = x; equivalently \sqrt[3]{x} = x^{1/3}

When to use: \sqrt[3]{27} asks: what number times itself times itself equals 27? Answer: 3, because 3 \times 3 \times 3 = 27. For negatives, \sqrt[3]{-8} = -2 because (-2) \times (-2) \times (-2) = -8.

Quick Example

\sqrt[3]{8} = 2 \quad \text{because } 2^3 = 8 \sqrt[3]{-27} = -3 \quad \text{because } (-3)^3 = -27 \sqrt[3]{64} = 4 \quad \text{because } 4^3 = 64

Notation

\sqrt[3]{x} is the cube root symbol; the small 3 distinguishes it from the square root \sqrt{x}

What This Formula Means

The cube root of x, written \sqrt[3]{x}, is the number that when multiplied by itself three times equals x. Unlike square roots, cube roots are defined for negative numbers.

\sqrt[3]{27} asks: what number times itself times itself equals 27? Answer: 3, because 3 \times 3 \times 3 = 27. For negatives, \sqrt[3]{-8} = -2 because (-2) \times (-2) \times (-2) = -8.

Formal View

\sqrt[3]{x} = x^{1/3} is the unique real number y such that y^3 = x. Defined for all x \in \mathbb{R} (unlike \sqrt{x}, which requires x \geq 0). Properties: \sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b} and \sqrt[3]{-x} = -\sqrt[3]{x}.

Worked Examples

Example 1

easy
Evaluate \sqrt[3]{-125}.

Solution

  1. 1
    A cube root asks for the number whose cube equals the expression inside the radical, so we want x^3 = -125.
  2. 2
    Test a likely integer: (-5)^3 = (-5)(-5)(-5) = 25 \times (-5) = -125.
  3. 3
    Therefore \sqrt[3]{-125} = -5.

Answer

-5
Unlike even roots, cube roots of negative numbers are real and negative. If a^3 = n, then \sqrt[3]{n} = a. This is because a negative times a negative times a negative is negative.

Example 2

medium
Simplify \sqrt[3]{54}.

Example 3

medium
Simplify \sqrt[3]{216x^6}.

Common Mistakes

  • Thinking cube roots can't be negative (they can: \sqrt[3]{-8} = -2)
  • Confusing cube roots with dividing by 3 (\sqrt[3]{27} = 3, not 27 \div 3 = 9)
  • Applying square root rules to cube roots (\sqrt[3]{a+b} \neq \sqrt[3]{a} + \sqrt[3]{b})

Why This Formula Matters

Cube roots appear when finding the side length of a cube from its volume, and in solving cubic equations.

Frequently Asked Questions

What is the Cube Roots formula?

The cube root of x, written \sqrt[3]{x}, is the number that when multiplied by itself three times equals x. Unlike square roots, cube roots are defined for negative numbers.

How do you use the Cube Roots formula?

\sqrt[3]{27} asks: what number times itself times itself equals 27? Answer: 3, because 3 \times 3 \times 3 = 27. For negatives, \sqrt[3]{-8} = -2 because (-2) \times (-2) \times (-2) = -8.

What do the symbols mean in the Cube Roots formula?

\sqrt[3]{x} is the cube root symbol; the small 3 distinguishes it from the square root \sqrt{x}

Why is the Cube Roots formula important in Math?

Cube roots appear when finding the side length of a cube from its volume, and in solving cubic equations.

What do students get wrong about Cube Roots?

Remembering that cube roots of negative numbers exist and are negative, unlike square roots which are not real for negative inputs.

What should I learn before the Cube Roots formula?

Before studying the Cube Roots formula, you should understand: exponents, square roots.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Cube Roots, Square Roots, and Irrational Numbers →
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