Calculate the cube root of any number — positive or negative — instantly. Get the exact decimal result, simplified radical form, and step-by-step workings. 100% free.
Racine cubique is the French word for cube root, and just like its cousin "racine carrée," it's a term people search in English all the time. If you landed here looking for a fast answer, the racine cubique calculator above will give you one instantly. Everything below explains what a cube root actually is, how to find one by hand, and where it shows up outside the classroom.
A cube root is a number that, when multiplied by itself three times, gives you another number. Multiply 3 by itself three times — 3 × 3 × 3 — and you get 27. So we say the cube root of 27 is 3.
That's the entire idea. Everything else on this page just builds on it.
Type your number into the box above.
Hit the Calculate button to process your number.
The calculator shows the cube root instantly, including a simplified form when one's available and correct handling of negative numbers.
No formulas needed to use the tool itself. The rest of this guide is for anyone who wants to understand what's happening behind that one click.
The symbol ∛ is the cube root sign. It's built from the same radical symbol used for square roots, but with a small "3" (called the index) tucked into the notch.
You'll also see cube roots written as a fractional exponent:
Both forms mean the same thing — the exponent version is often used in calculators, spreadsheets, and programming because the ∛ symbol isn't always easy to type.
Like square roots, there's no single plug-and-play formula — a cube root is defined by a relationship:
In plain words: whatever answer you get, multiplying it by itself three times has to bring you back to your starting number.
These five rules mirror the square root laws you may already know — but adapted for the third degree. Once you master them, the nth root calculator generalizes the same patterns even further.
Example: ∛8 × ∛27 = 2 × 3 = 6, and ∛(8×27) = ∛216 = 6. Same answer.
Just like with square roots (racine carrée), you can break a complicated cube root into simpler pieces. Instead of computing ∛216 directly, factor it as 8 × 27, compute each cube root separately, and multiply. This is the go-to technique for simplifying expressions like ∛(24) = ∛(8×3) = 2∛3.
Example: ∛64 ÷ ∛8 = 4 ÷ 2 = 2, and ∛(64÷8) = ∛8 = 2.
The mirror of the product rule — take the cube root of a fraction by rooting the numerator and denominator separately. Particularly useful when simplifying algebraic expressions like ∛(125/8) = ∛125 ÷ ∛8 = 5/2.
Example: ∛(5³) = ∛125 = 5. Cubing and cube-rooting undo each other.
Unlike square roots, which only return non-negative values, the power rule for cube roots works for all real numbers — positive, negative, and zero. ∛((-3)³) = ∛(-27) = -3. This is a key difference from the racine carrée (which needs absolute value for negatives).
Example: (∛5)³ = 5. (∛12)³ = 12. It always works.
This is the definition of cube root stated in reverse. If ∛a is "the number that, when cubed, gives a," then cubing ∛a must give you a back. This identity is used to rationalize cube-root denominators and appears in solving cubic equations.
Example: No matter the context, ∛1 is always 1 and ∛0 is always 0.
These are the fixed points of the cube root function. ∛1 = 1 because 1³ = 1, and ∛0 = 0 because 0³ = 0. They serve as boundary conditions in proofs and as sanity checks for formulas — plug in 0 or 1, and you should always get a sensible answer.
Type in your number, click Calculate, done. This is the method most people reach for once they know it exists — especially for decimals or large numbers where hand methods get slow. You can also use our square root calculator for second-degree roots or the nth root calculator for any custom degree.
This method breaks a number into its prime factors and groups them in sets of three.
This is an approximation method used by computers and scientific calculators to home in on a very precise decimal answer through repeated refinement.
Every positive number has exactly one real cube root, and it's always positive.
| Number | Cube Root | Why |
|---|---|---|
| 8 | 2 | 2 × 2 × 2 = 8 |
| 125 | 5 | 5 × 5 × 5 = 125 |
| 512 | 8 | 8 × 8 × 8 = 512 |
| 10 | ≈2.154 | Not a whole number, but still a real, exact value |
Unlike square roots, cube roots don't have a "second" answer to worry about — there's only one real cube root for any given number.
Here's the single most important thing that separates cube roots from square roots: you can take the cube root of a negative number, and the answer is a normal, real number.
Why does this work when it doesn't for square roots? Because multiplying three negative numbers together gives you a negative result: negative × negative = positive, then positive × negative = negative.
That's the full answer. No "i," no imaginary unit, nothing extra needed.
Square roots of negative numbers require the imaginary unit i, but cube roots handle negatives just fine on their own. This is genuinely useful to know if you've also looked at our Racine Carrée Calculator.
| Cube Root | Square Root | |
|---|---|---|
| Symbol | ∛ | √ |
| Multiplications | 3 (x × x × x) | 2 (x × x) |
| Example | ∛64 = 4 (4×4×4=64) | √64 = 8 (8×8=64) |
| Negative numbers | Real answer exists (∛-64 = -4) | No real answer (needs "i") |
If you need a square root specifically, our Racine Carrée Calculator covers that in full, including the imaginary number rules.
While square roots (racine carrée) deal with two-dimensional problems, cube roots shine whenever the problem involves three dimensions:
Architects and designers use cube roots to find the edge length of cubic spaces from a known volume. If a client wants a 64 m³ storage room in a cube shape, ∛64 = 4 m per side. This applies to water tanks, HVAC ducts, and any three-dimensional space planning.
Logistics companies calculate box dimensions from a target volume using cube roots. If a parcel must fit exactly 1,000 cm³, the cube root tells them each side needs to be ∛1000 = 10 cm. This feeds directly into shipping cost formulas that use "dimensional weight."
Volume-related formulas in fluid dynamics, thermodynamics, and materials science frequently involve cube roots. For instance, the radius of a sphere from its volume (r = ∛(3V/4π)) is a cube root operation. Scaling laws also use cube roots — doubling the volume of a structure doesn't double its dimensions.
When you know a total return over three years and want the annual compound growth rate, you need a cube root. If an investment grows from $1,000 to $1,331 in three years, the annual rate is ∛(1.331) - 1 ≈ 10%. This concept extends to nth roots for longer periods.
If you know how much a cube-shaped container holds — say, a 27-liter fish tank — the cube root tells you the dimensions: ∛27 = 3, so each edge is roughly 30 cm. No ruler needed, just the racine cubique.
Not every cube root is a perfect cube, but many can still be simplified by pulling out any perfect-cube factors.
| Expression | Simplified Form | Why |
|---|---|---|
| ∛16 | 2∛2 | 16 = 8 × 2, and ∛8 = 2 |
| ∛24 | 2∛3 | 24 = 8 × 3 |
| ∛40 | 2∛5 | 40 = 8 × 5 |
| ∛54 | 3∛2 | 54 = 27 × 2 |
| ∛81 | 3∛3 | 81 = 27 × 3 |
| ∛128 | 4∛2 | 128 = 64 × 2 |
| ∛250 | 5∛2 | 250 = 125 × 2 |
The trick: find the largest perfect cube that divides evenly into your number, pull it out, and the rest stays under the radical.
Bookmark this section for quick lookups. Perfect cubes are highlighted in bold with a badge. Values are rounded to three decimal places. For square root reference tables, see our main homepage.
| Number (n) | Cube Root (∛n) | Type |
|---|---|---|
| 1 | 1.000 | Perfect Cube |
| 2 | 1.260 | Irrational |
| 3 | 1.442 | Irrational |
| 4 | 1.587 | Irrational |
| 5 | 1.710 | Irrational |
| 6 | 1.817 | Irrational |
| 7 | 1.913 | Irrational |
| 8 | 2.000 | Perfect Cube |
| 9 | 2.080 | Irrational |
| 10 | 2.154 | Irrational |
| 11 | 2.224 | Irrational |
| 12 | 2.289 | Irrational |
| 13 | 2.351 | Irrational |
| 14 | 2.410 | Irrational |
| 15 | 2.466 | Irrational |
| 16 | 2.520 | Irrational |
| 17 | 2.571 | Irrational |
| 18 | 2.621 | Irrational |
| 19 | 2.668 | Irrational |
| 20 | 2.714 | Irrational |
| 21 | 2.759 | Irrational |
| 22 | 2.802 | Irrational |
| 23 | 2.844 | Irrational |
| 24 | 2.884 | Irrational |
| 25 | 2.924 | Irrational |
| 26 | 2.962 | Irrational |
| 27 | 3.000 | Perfect Cube |
| 28 | 3.037 | Irrational |
| 29 | 3.072 | Irrational |
| 30 | 3.107 | Irrational |
| 31 | 3.141 | Irrational |
| 32 | 3.175 | Irrational |
| 33 | 3.208 | Irrational |
| 34 | 3.240 | Irrational |
| 35 | 3.271 | Irrational |
| 36 | 3.302 | Irrational |
| 37 | 3.332 | Irrational |
| 38 | 3.362 | Irrational |
| 39 | 3.391 | Irrational |
| 40 | 3.420 | Irrational |
| 41 | 3.448 | Irrational |
| 42 | 3.476 | Irrational |
| 43 | 3.503 | Irrational |
| 44 | 3.530 | Irrational |
| 45 | 3.557 | Irrational |
| 46 | 3.583 | Irrational |
| 47 | 3.609 | Irrational |
| 48 | 3.634 | Irrational |
| 49 | 3.659 | Irrational |
| 50 | 3.684 | Irrational |
| 51 | 3.708 | Irrational |
| 52 | 3.733 | Irrational |
| 53 | 3.756 | Irrational |
| 54 | 3.780 | Irrational |
| 55 | 3.803 | Irrational |
| 56 | 3.826 | Irrational |
| 57 | 3.849 | Irrational |
| 58 | 3.871 | Irrational |
| 59 | 3.893 | Irrational |
| 60 | 3.915 | Irrational |
| 61 | 3.936 | Irrational |
| 62 | 3.958 | Irrational |
| 63 | 3.979 | Irrational |
| 64 | 4.000 | Perfect Cube |
| 65 | 4.021 | Irrational |
| 66 | 4.041 | Irrational |
| 67 | 4.062 | Irrational |
| 68 | 4.082 | Irrational |
| 69 | 4.102 | Irrational |
| 70 | 4.121 | Irrational |
| 71 | 4.141 | Irrational |
| 72 | 4.160 | Irrational |
| 73 | 4.179 | Irrational |
| 74 | 4.198 | Irrational |
| 75 | 4.217 | Irrational |
| 76 | 4.236 | Irrational |
| 77 | 4.254 | Irrational |
| 78 | 4.273 | Irrational |
| 79 | 4.291 | Irrational |
| 80 | 4.309 | Irrational |
| 81 | 4.327 | Irrational |
| 82 | 4.344 | Irrational |
| 83 | 4.362 | Irrational |
| 84 | 4.380 | Irrational |
| 85 | 4.397 | Irrational |
| 86 | 4.414 | Irrational |
| 87 | 4.431 | Irrational |
| 88 | 4.448 | Irrational |
| 89 | 4.465 | Irrational |
| 90 | 4.481 | Irrational |
| 91 | 4.498 | Irrational |
| 92 | 4.514 | Irrational |
| 93 | 4.531 | Irrational |
| 94 | 4.547 | Irrational |
| 95 | 4.563 | Irrational |
| 96 | 4.579 | Irrational |
| 97 | 4.595 | Irrational |
| 98 | 4.610 | Irrational |
| 99 | 4.626 | Irrational |
| 100 | 4.642 | Irrational |
A perfect cube is a number you get by multiplying a whole number by itself three times. Cubes grow much faster than squares, which is why this list stops at 10.
| Number | Cube | Number | Cube |
|---|---|---|---|
| 1 | 1 | 6 | 216 |
| 2 | 8 | 7 | 343 |
| 3 | 27 | 8 | 512 |
| 4 | 64 | 9 | 729 |
| 5 | 125 | 10 | 1,000 |
Type any number into the box above and click Calculate. The tool instantly returns the cube root, including a simplified form where one exists, and it correctly handles negative numbers as real values.
Yes. For perfect cubes, prime factorization works cleanly. For everything else, estimation (finding which two perfect cubes your number falls between) gets you close fast.
It's French for "cube root." A racine cubique calculator and a cube root calculator do exactly the same job.
Yes, and this is one of the biggest differences between cube roots and square roots. The cube root of a negative number is simply a negative real number — for example, ∛-8 = -2. No imaginary numbers involved.
Yes. The calculator handles decimals, whole numbers, and large values with full precision.
A square root asks what number times itself equals your value. A cube root asks what number times itself three times equals your value. See the comparison table above for a full breakdown.
Yes — no sign-up, no downloads, no usage limits.
A cube root answers one question: what number, multiplied by itself three times, gives you this value? That single idea carries you through everything else on this page — notation, formulas, manual methods, and the fact that (unlike square roots) cube roots handle negative numbers without needing anything imaginary. Use the calculator above for a fast, exact answer, and come back here anytime you want to work through the math yourself.