Cube Roots Formula

Cube roots are the cube root [3]x is the number that, when cubed, gives x — defined for all real numbers, including negatives.

The Formula

x3=n   means   n3=x\sqrt[3]{x}=n\;\text{ means }\;n^3=x

When to use: 273\sqrt[3]{27} asks: what number times itself times itself equals 27? Answer: 3, because 3×3×3=273 \times 3 \times 3 = 27. For negatives, −83=−2\sqrt[3]{-8} = -2 because (−2)×(−2)×(−2)=−8(-2) \times (-2) \times (-2) = -8.

Quick Example

83=2because 23=8\sqrt[3]{8} = 2 \quad \text{because } 2^3 = 8 −273=−3because (−3)3=−27\sqrt[3]{-27} = -3 \quad \text{because } (-3)^3 = -27 643=4because 43=64\sqrt[3]{64} = 4 \quad \text{because } 4^3 = 64

Notation

A cube root asks which number multiplied by itself three times gives the radicand.

What This Formula Means

The cube root x3\sqrt[3]{x} is the number that, when cubed, gives xx — defined for all real numbers, including negatives.

273\sqrt[3]{27} asks: what number times itself times itself equals 27? Answer: 3, because 3×3×3=273 \times 3 \times 3 = 27. For negatives, −83=−2\sqrt[3]{-8} = -2 because (−2)×(−2)×(−2)=−8(-2) \times (-2) \times (-2) = -8.

Formal View

x3=x1/3\sqrt[3]{x} = x^{1/3} is the unique real number yy such that y3=xy^3 = x. Defined for all x∈Rx \in \mathbb{R} (unlike x\sqrt{x}, which requires x≥0x \geq 0). Properties: ab3=a3 b3\sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b} and −x3=−x3\sqrt[3]{-x} = -\sqrt[3]{x}.

Worked Examples

Example 1

easy
Evaluate −1253\sqrt[3]{-125}.

Answer

−5-5

First step

1
A cube root asks for the number whose cube equals the expression inside the radical, so we want x3=−125x^3 = -125.

Full solution

  1. 2
    Test a likely integer: (−5)3=(−5)(−5)(−5)=25×(−5)=−125(-5)^3 = (-5)(-5)(-5) = 25 \times (-5) = -125.
  2. 3
    Therefore −1253=−5\sqrt[3]{-125} = -5.
Unlike even roots, cube roots of negative numbers are real and negative. If a3=na^3 = n, then n3=a\sqrt[3]{n} = a. This is because a negative times a negative times a negative is negative.

Example 2

medium
Simplify 543\sqrt[3]{54}.

Example 3

medium
Simplify 216x63\sqrt[3]{216x^6}.

Common Mistakes

  • Using square root because the radical looks familiar — check the small index 3.
  • Dividing by 3 instead of undoing a cube — 273=3\sqrt[3]{27}=3, not 9.
  • Forgetting negative cube roots can be negative — −83=−2\sqrt[3]{-8}=-2.

Why This Formula Matters

Cube roots extend inverse operations beyond squares and help students distinguish area thinking from volume thinking. Recognizing it by "What number multiplied by itself three times gives this value?" — rather than by familiar numbers — is what lets a student tell it apart from square roots and exponents in a mixed problem set.

Frequently Asked Questions

What is the Cube Roots formula?

The cube root x3\sqrt[3]{x} is the number that, when cubed, gives xx — defined for all real numbers, including negatives.

How do you use the Cube Roots formula?

273\sqrt[3]{27} asks: what number times itself times itself equals 27? Answer: 3, because 3×3×3=273 \times 3 \times 3 = 27. For negatives, −83=−2\sqrt[3]{-8} = -2 because (−2)×(−2)×(−2)=−8(-2) \times (-2) \times (-2) = -8.

What do the symbols mean in the Cube Roots formula?

A cube root asks which number multiplied by itself three times gives the radicand.

Why is the Cube Roots formula important in Math?

Cube roots extend inverse operations beyond squares and help students distinguish area thinking from volume thinking. Recognizing it by "What number multiplied by itself three times gives this value?" — rather than by familiar numbers — is what lets a student tell it apart from square roots and exponents in a mixed problem set.

What do students get wrong about Cube Roots?

The procedure for cube roots is the easy part; the trap is using square root because the radical looks familiar. Asking "What number multiplied by itself three times gives this value?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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 →
← Back to Cube Roots See All Examples Practice Problems