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Cube Root — Definition, Formula & Examples

Cube Root

A number that must be multiplied times itself three times to equal a given number. The cube root of x is written Cube root of x written as the radical symbol with a small 3 as the index: ∛xor x to the power of 1/3, representing the cube root of x using rational exponent notation.

For example, Cube root of 8 equals 2, written as ∛8 = 2 since 2 · 2 · 2 = 2³ = 8, showing that 2 multiplied by itself three times equals 8, illustrating a cube root example..

 

 

See also

Square root, nth root, rational exponents, radical rules

Key Formula

x3=x1/3\sqrt[3]{x} = x^{1/3}
Where:
  • xx = The number whose cube root you want to find
  • x3\sqrt[3]{x} = The cube root of x, meaning the value a such that a³ = x

Worked Example

Problem: Find the cube root of 64.
Step 1: Ask: what number multiplied by itself three times equals 64?
a3=64a^3 = 64
Step 2: Test a = 4. Compute 4 × 4 × 4.
43=4×4×4=644^3 = 4 \times 4 \times 4 = 64
Step 3: Since 4³ = 64, the cube root of 64 is 4.
643=4\sqrt[3]{64} = 4
Answer: The cube root of 64 is 4.

Another Example

Problem: Find the cube root of −125.
Step 1: Ask: what number cubed gives −125?
a3=−125a^3 = -125
Step 2: You know that 5³ = 125. Since a negative number cubed stays negative, try a = −5.
(−5)3=(−5)×(−5)×(−5)=−125(-5)^3 = (-5) \times (-5) \times (-5) = -125
Step 3: Confirm the result.
−1253=−5\sqrt[3]{-125} = -5
Answer: The cube root of −125 is −5.

Frequently Asked Questions

Can you take the cube root of a negative number?
Yes. Unlike square roots, cube roots of negative numbers are real. A negative number multiplied by itself three times gives a negative result, so every negative number has a real cube root. For example, ∛(−8) = −2 because (−2)³ = −8.
What is the difference between cube root and cubing a number?
Cubing a number means raising it to the third power (multiplying it by itself three times), like 5³ = 125. Taking the cube root is the inverse operation — it asks which number was cubed to get a given result, like ∛125 = 5. They undo each other.

Cube Root vs. Square Root

A square root finds which number squared equals the input (√16 = 4 because 4² = 16), while a cube root finds which number cubed equals the input (∛27 = 3 because 3³ = 27). A key difference: square roots of negative numbers are not real, but cube roots of negative numbers are always real. Additionally, the principal square root is always non-negative, whereas a cube root preserves the sign of the original number.

Why It Matters

Cube roots appear whenever you reverse a cubing operation, such as finding the side length of a cube when you know its volume. If a cube has a volume of 216 cubic centimeters, its side length is ∛216 = 6 cm. Cube roots also show up in science and engineering formulas, including those for scaling, fluid dynamics, and material strength.

Common Mistakes

Mistake: Thinking the cube root of a negative number is undefined or "not real."
Correction: That rule applies to even roots (like square roots), not odd roots. A negative number cubed is negative, so its cube root is a real negative number. For example, ∛(−27) = −3.
Mistake: Confusing cube root with dividing by 3.
Correction: Dividing 27 by 3 gives 9, but ∛27 = 3 because 3 × 3 × 3 = 27. The cube root asks for the number that, when used as a factor three times, produces the input — it is not simple division.

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