Key takeaways
- ∛n is the value that, multiplied by itself three times, gives n.
- Check a perfect cube first: 1, 8, 27, 64, 125... have whole-number cube roots.
- Unlike square roots, negative numbers DO have real cube roots.
- Cubing your answer should give back the original number.
In this guide
What is a cube root?
The cube root of a number n is the value that, multiplied by itself three times, gives n. For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27. The cube root symbol is ∛.
Why does the cube root matter?
Cube roots undo cubing, which is exactly what happens when you calculate volume. Whenever you know a volume and need to find the side length of a cube, a cube root gets you there.
Unlike square roots, a negative number DOES have a real cube root:∛−8 = −2, because−2 × −2 × −2 = −8. A negative times itself three times stays negative.
The formula
| Symbol | What it means | Example |
|---|---|---|
| ∛ | The cube root symbol | — |
| n | The number you're finding the cube root of | 512 |
How to find a cube root step by step
- 1Check if the number is a perfect cube, meaning a whole number multiplied by itself three times equals it (like 1, 8, 27, 64, 125...).
- 2If it is, find the whole number that, cubed, gives your number.
- 3If it isn't a perfect cube, estimate between the two nearest perfect cubes, or use a calculator for a decimal answer.
- 4Check your answer by cubing it (multiplying it by itself three times) — it should equal the original number.
Enter your own number in the calculator below to see the check step laid out.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting that negative numbers do have real cube roots. Unlike square roots, the cube root of a negative number is negative.
- Mixing up units. If n is a volume, like 512 cm³, the cube root gives a length, like 8 cm, not a cubed unit.
- Guessing without checking. Always cube your estimate to confirm it matches the original number.
Tips and tricks
- Estimate a non-perfect cube by finding the two nearest perfect cubes and judging which one it's closer to.
- Double-check any decimal answer by cubing it — it should land close to the original number.
- Cube roots reverse exponents, and for flat shapes instead of solids, see the square root calculator.
Where you'll use it in real life
- Packing and storage: finding the side length of a cube-shaped box from its volume.
- Construction: working backward from a known volume to a container's dimensions, using the cube's volume formula.
- Science: scaling relationships where a quantity depends on the cube of a length, like volume or weight.
- Math class: solving equations that involve a variable raised to the third power.
Quick summary
To find a cube root, check whether the number is a perfect cube, then find (or estimate) the value that, cubed, gives it back. Unlike square roots, cube roots of negative numbers are perfectly valid. Try the calculator above with your own number.
Worked example
A cube-shaped storage box has a volume of 512 cubic centimeters. How long is each side of the box?
- 01∛512
- 028 × 8 × 8 = 512
- 03∛512 = 8
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the difference between a square root and a cube root?
A square root undoes squaring (multiplying a number by itself twice), while a cube root undoes cubing (multiplying a number by itself three times). They usually give different answers for the same input.
Can a negative number have a cube root?
Yes. Unlike square roots, cube roots of negative numbers are real. For example, the cube root of −27 is −3, because −3 × −3 × −3 = −27.
How do I find the cube root of a number that isn't a perfect cube?
Estimate between the two nearest perfect cubes, then refine with a calculator. For example, ∛50 is between ∛27 (3) and ∛64 (4), closer to 3.68.
What is the cube root of 1?
It's 1, because 1 × 1 × 1 = 1. In fact, both the square root and cube root of 1 are always 1.
