Can you take the cube root of a negative number?

Yes, and the result is negative. For example, \sqrt[3]{-27} = -3 because (-3)^3 = -27. Cube roots are unique for every real number, including negatives, which makes them different from square roots.

What's the difference between \sqrt[3]{8} and \sqrt{8}?

\sqrt[3]{8} is the cube root (the number you cube to get 8), which is 2. Meanwhile, \sqrt{8} is the square root (the number you square to get 8), which is about 2.83. The small 3 in \sqrt[3]{x} tells you it's cube root, not square root.

How do I simplify \sqrt[3]{40}?

Factor 40 as 8 \times 5. Since \sqrt[3]{8} = 2, you get \sqrt[3]{40} = \sqrt[3]{8 \times 5} = \sqrt[3]{8} \cdot \sqrt[3]{5} = 2\sqrt[3]{5}. Use prime factorization to find perfect cubes you can pull out.

Why do we even need cube root if we can just use a calculator?

Calculators are useful, but understanding cube root helps you estimate answers, simplify expressions in algebra, and solve equations by hand. It's a fundamental operation that appears in volume, physics, and higher mathematics.

Cube Root

Cube root finds the number that, when multiplied by itself three times, equals your original number—the inverse of cubing.

\sqrt[3]{a} = b \iff b^3 = a

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Type the problem. The solver will use Cube Root where Cube Root is the right tool, and tell you when it is not.

What each symbol means

What Cube Root takes

Symbol	Meaning
$a$	The number under the cube root symbol (the radicand); it can be any real number including negatives, and represents the starting value in $\sqrt[3]{a} = b$ .
$b$	The answer to the cube root operation—the number that, when multiplied by itself three times, gives $a$ ; it's the value of $b$ in the equation $b^3 = a$ .

When to use it

Use cube root when you need to find what number, multiplied by itself three times, gives you a known result.

Level

Usually taught in: Pre-Algebra

Worked examples

1. Find the cube root of a perfect cube

Problem

Find

\sqrt[3]{27}

$b^3 = 27$
We need to find the value of $b$ where $b$ multiplied by itself three times equals 27.
$3^3 = ?$
Let's test $b = 3$ by cubing it.
$3^3 = 3 \cdot 3 \cdot 3 = 9 \cdot 3 = 27$
Multiply $3 \times 3 = 9$ , then $9 \times 3 = 27$ , so this works.

Answer: $\sqrt[3]{27} = 3$

We found the number that makes $b^3 = 27$ true. Since $3^3 = 27$ , the cube root of 27 is 3.

2. Find the cube root of a negative number

Problem

Find

\sqrt[3]{-8}

$b^3 = -8$
We need to find $b$ such that $b$ multiplied by itself three times equals -8.
$(-2)^3 = ?$
Let's test $b = -2$ by cubing it.
$(-2)^3 = (-2) \cdot (-2) \cdot (-2) = 4 \cdot (-2) = -8$
Multiply $(-2) \times (-2) = 4$ (negative times negative is positive), then $4 \times (-2) = -8$ (positive times negative is negative).

Answer: $\sqrt[3]{-8} = -2$

Cube roots of negative numbers are real and negative. Since $(-2)^3 = -8$ , the cube root of -8 is -2.

3. Use cube root to solve a video game level problem

Problem

In a video game, you earn experience points that scale with your level cubed. At level

n

, you've earned

n^3 \times 1000

total points. If you've earned 8,000 points, what level are you?

$n^3 \cdot 1000 = 8000$
The total points equation, where $n$ is the current level.
$n^3 = \frac{8000}{1000} = 8$
Divide both sides by 1000 to isolate $n^3$ .
$n = \sqrt[3]{8}$
Take the cube root of both sides to find the level.
$2^3 = 2 \cdot 2 \cdot 2 = 4 \cdot 2 = 8$
Check: $2 \times 2 \times 2 = 4 \times 2 = 8$ , so $\sqrt[3]{8} = 2$ .

Answer: $n = 2$

By isolating $n^3$ and taking the cube root, we found that you're at level 2. This shows how cube root helps reverse the cubing operation in real problems.

Common mistakes

Where Cube Root usually goes wrong

The mistake	Why it is wrong	The fix
Confusing $\sqrt[3]{a}$ with $\sqrt{a}$ (the square root)	Cube root and square root are different operations—cube root uses three factors, square root uses two.	Remember the small 3 in the radical symbol: $\sqrt[3]{x}$ means cube root, but $\sqrt{x}$ means square root. The cube root of 8 is 2 (since $2^3 = 8$ ), but the square root of 8 is about 2.83 (since $2.83^2 \approx 8$ ).
Thinking that cube roots of negative numbers don't exist or are undefined	Cube root of a negative number is a real negative number, not undefined or impossible.	Cube roots of negative numbers are allowed and give negative answers. For example, $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$ . The rule is: if $a$ is negative, then $\sqrt[3]{a}$ is negative.
Skipping the verification step and not checking your answer by cubing it	Without verification, arithmetic errors go unnoticed and lead to confident but wrong answers.	Always check: if you say $\sqrt[3]{a} = b$ , then cube $b$ and verify you get $a$ . For $\sqrt[3]{27} = 3$ , check: $3^3 = 27$ ✓.

Tips and when to use something else

Memorize the cubes of small numbers (1–10) to spot cube roots instantly: $1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125$ , etc.
Unlike square root, cube root works with negative numbers: $\sqrt[3]{-a} = -\sqrt[3]{a}$ . This is because a negative times a negative times a negative is negative.
If a cube root doesn't simplify to an integer, leave it in radical form (like $2\sqrt[3]{10}$ ) rather than rounding, unless the problem asks for a decimal approximation.
For larger numbers, use Prime Factorization to break them down: if $a = p_1^3 \cdot p_2^3 \cdot p_3 \cdot \ldots$ , then $\sqrt[3]{a} = p_1 \cdot p_2 \cdot \sqrt[3]{p_3 \cdot \ldots}$ .

Related formulas

Square RootSquare root finds the non-negative number that, when multiplied by itself, gives your starting number; use it to undo squaring in equations.Prime FactorizationPrime factorization breaks any whole number into a unique multiplication of primes—essential for finding common factors and simplifying fractions.Order of OperationsOrder of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.Rules of SignsRules of Signs tell you whether the product of two numbers is positive or negative, which you need for solving equations with negative numbers.

Frequently asked questions

Can you take the cube root of a negative number?: Yes, and the result is negative. For example, $\sqrt[3]{-27} = -3$ because $(-3)^3 = -27$ . Cube roots are unique for every real number, including negatives, which makes them different from square roots.
What's the difference between $\sqrt[3]{8}$ and $\sqrt{8}$ ?: $\sqrt[3]{8}$ is the cube root (the number you cube to get 8), which is 2. Meanwhile, $\sqrt{8}$ is the square root (the number you square to get 8), which is about 2.83. The small 3 in $\sqrt[3]{x}$ tells you it's cube root, not square root.
How do I simplify $\sqrt[3]{40}$ ?: Factor 40 as $8 \times 5$ . Since $\sqrt[3]{8} = 2$ , you get $\sqrt[3]{40} = \sqrt[3]{8 \times 5} = \sqrt[3]{8} \cdot \sqrt[3]{5} = 2\sqrt[3]{5}$ . Use prime factorization to find perfect cubes you can pull out.
Why do we even need cube root if we can just use a calculator?: Calculators are useful, but understanding cube root helps you estimate answers, simplify expressions in algebra, and solve equations by hand. It's a fundamental operation that appears in volume, physics, and higher mathematics.

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Reviewed 2026-09-18