Cube Roots Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Cube Roots.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

$\sqrt[3]{27}$ 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 $(-2) \times (-2) \times (-2) = -8$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A cube root asks for the edge length of a cube with a given volume.

Common stuck point: 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.

Sense of Study hint: Ask: What number multiplied by itself three times gives this value?

Worked Examples

Example 1

easy

Evaluate

\sqrt[3]{-125}

Answer

-5

First step

A cube root asks for the number whose cube equals the expression inside the radical, so we want

x^3 = -125

Full solution

2
Test a likely integer: $(-5)^3 = (-5)(-5)(-5) = 25 \times (-5) = -125$ .
3
Therefore $\sqrt[3]{-125} = -5$ .

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}

Example 4

medium

Simplify

\sqrt[3]{x^9}

for

x > 0

Example 5

medium

Simplify

\sqrt[3]{27a^3}

for any real

a

Example 6

hard

Simplify

\sqrt[3]{54a^4}

for

a > 0

Example 7

hard

Solve

x^3 + 8 = 0

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Evaluate

\sqrt[3]{8} + \sqrt[3]{27}

Example 2

easy

Evaluate

\sqrt[3]{-64} + \sqrt[3]{216}

Example 3

easy

Find

\sqrt[3]{27}

Example 4

easy

Find

\sqrt[3]{8}

Example 5

easy

Find

\sqrt[3]{-8}

Example 6

easy

Find

\sqrt[3]{1}

Example 7

easy

Find

\sqrt[3]{64}

Example 8

easy

Find

\sqrt[3]{1000}

Example 9

easy

Find

\sqrt[3]{0}

Example 10

easy

\sqrt[3]{125}

rational?

Example 11

medium

Solve

x^3=27

for real

x

Example 12

medium

Solve

x^3=-64

for real

x

Example 13

medium

Simplify

\sqrt[3]{54}

Example 14

medium

\sqrt[3]{20}

rational or irrational?

Example 15

medium

Find

\sqrt[3]{27}\cdot\sqrt[3]{8}

Example 16

medium

Between which two integers does

\sqrt[3]{50}

lie?

Example 17

medium

A cube has volume

125\,\text{cm}^3

. Find its edge length.

Example 18

challenge

Solve

2x^3=54

for real

x

Example 19

challenge

Simplify

\sqrt[3]{16}+\sqrt[3]{54}

Example 20

challenge

Find

\sqrt[3]{\sqrt{64}}

Example 21

medium

Solve

x^3=8

and explain why there is one real solution.

Example 22

medium

Simplify

\sqrt[3]{24}

Example 23

easy

Find

\sqrt[3]{125}

Example 24

easy

Find

\sqrt[3]{-27}

Example 25

easy

A cube has volume

343

cubic units. What is the edge length?

Example 26

easy

Evaluate

\sqrt[3]{512}

Example 27

easy

Evaluate

\sqrt[3]{-216}

Example 28

medium

Simplify

\sqrt[3]{16}

Example 29

medium

Simplify

\sqrt[3]{40}

Example 30

medium

Simplify

\sqrt[3]{-128}

Example 31

medium

Solve

x^3 = 64

Example 32

medium

Solve

x^3 = -125

Example 33

medium

Estimate

\sqrt[3]{50}

to the nearest tenth.

Example 34

medium

Simplify

\sqrt[3]{125x^6 y^3}

Example 35

hard

A cube has volume

1331

^3

. Find its surface area.

Example 36

hard

Solve

2x^3 = 250

Example 37

hard

Solve

(x-1)^3 = 27

Example 38

hard

Find

\sqrt[3]{0.008}

Example 39

hard

Compute

\sqrt[3]{8} \cdot \sqrt[3]{27}

Example 40

challenge

A cube of edge

a

has the same volume as a rectangular box of dimensions

2 \times 4 \times 27

. Find

a

← Back to Cube Roots Formula Explained Practice Mode

Related Concepts

Exponents Square Roots Irrational Numbers

Background Knowledge

These ideas may be useful before you work through the harder examples.

exponentssquare roots