Perfect Cubes And Cube Roots

What Is A Cube?

A cube is a number multiplied by itself three times:
a³ = a × a × a
Examples:
2³ = 2 × 2 × 2 = 8
(-3)³ = -3 × -3 × -3 = -27

What Is A Cube Root?

Finding the cube root is the inverse operation of cubing.
The cube root of a number a is a value x such that:
x³ = a
It is written as ∛a = x
Cube root works for all real numbers (positive, negative, or zero).

∛ is the special symbol that means “cube root”. It is the radical symbol √ with a little three to mean cube root.
You can use it like this:
∛27 = 3.

Examples:
∛8 = 2 (since 3³ = 8)
∛(-27) = -3 (since (-3)³ = 27
∛0 = 0

What Is A Perfect Cube?

A perfect cube is an integer that is the cube of another integer:
n = k3³ (where k is an integer)

For example, 125 is a perfect cube since 125 = 5 × 5 × 5 = 5³
Some other examples of perfect cubes are 1, 8, 27, 64, 125, 216, 343, …

The following table shows perfect cubes from 1 to 1000 and their corresponding cube roots. Scroll down the page for more examples of how to evaluate cube root of perfect cubes.

 

It is easy to work out the cube root of a perfect cube, but it is hard to work out other cube roots.

Exponent Worksheets
Practice your skills with the following exponent worksheets:
Printable & Online Exponent Worksheets

How to Find Cube Roots
Method 1: Memorization (for Small Numbers)
Memorize cubes of integers 1–10 (see table above).

Method 2: Prime Factorization

1. Factor the number into primes.
   
2. Group triples of identical factors.
   
3. Multiply one factor from each triple.
   

Example: Find ∛216

1. Prime factors: 216 = 2 × 2 × 2 × 3 × 3 × 3
   
2. Triples: (2 × 2 × 2) and (3 × 3 × 3)
   
3. Cube root: 2 × 3 = 6
   

Perfect Cubes & Cube Roots

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Evaluate Square Roots Of Perfect Squares And Cube Roots Of Perfect Cubes

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How To Simplify Radicals With Negative Radicands And Odd Indexes?

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How To Find The Smallest Multiple Of Another Number That Is Also A Perfect Cube?

Example:
What is the smallest positive whole number that is a perfect cube and is a multiple of 24?

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Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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