Racine Cubique Calculator | Free Online Cube Root Calculator

Complete Guide

What Is Racine Cubique (Cube Root)?

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.

Radical Anatomy Explorer Click or hover over any part

³ √ 27 = 3

Index (Degree 3)

The small number 3 in the upper-left of the radical sign. It indicates you are finding the 3rd root (cube root) rather than a square root (index 2).

Radical Symbol

The mathematical sign ∛ denoting the extraction of the cube root. In Unicode, it is character U+221B.

Radicand

The number underneath the radical sign whose cube root is being calculated. It can be positive, negative, or zero.

Cube Root (Result)

The real value that, when multiplied by itself three times (y × y × y), equals the radicand.

∛8 = 2 because 2 × 2 × 2 = 8

∛64 = 4 because 4 × 4 × 4 = 64

∛1000 = 10 because 10 × 10 × 10 = 1,000

That's the entire idea. Everything else on this page just builds on it.

How To Use

How to Use the Racine Cubique Calculator

Get your answer in three simple steps:

1

Type Your Number

Type your number into the box above.

2

Click Calculate

Hit the Calculate button to process your number.

3

Read Your Result

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.

Notation

Cube Root Notation: What Does the ∛ Symbol Mean?

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.

∛64 = 4 Copy

Standard mathematical radical notation using index 3 and radical sign.

64^(1/3) = 4 Copy

Fractional power representation: raising any number to 1/3 equals its cube root.

Math.cbrt(64) // 4 Copy

Standard in JavaScript, Python (64 ** (1/3)), C, Java, and modern runtimes.

=POWER(A1, 1/3) Copy

Excel and Google Sheets formula. Alternatively use =A1^(1/3).

\sqrt[3]64 Copy

LaTeX mathematical typesetting command for academic papers and documents.

Unicode: U+221B | HTML: &#8731; | 221B + Alt+X Copy ∛

On Windows: type 221B then press Alt+X. On Mac: Cmd+Ctrl+Space. HTML: ∛.

You'll also see cube roots written as a fractional exponent:

x^(1/3) = ∛x

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.

So whether you write ∛x or x^(1/3), the operation is identical.

Formula

What Is the Formula for Cube Roots?

Like square roots, there's no single plug-and-play formula: a cube root is defined by a relationship:

If y = ∛x, then y \times y \times y = x (or y³ = x) Copy

In plain words: whatever answer you get, multiplying it by itself three times has to bring you back to your starting number.

Core Laws

The Five Core Laws of Cube Roots

These five rules mirror the foundational square root rules you may already know, but adapted for the third degree:

1

Product Rule

Split products inside radicals

Example: ∛8 \times ∛27 = 2 \times 3 = 6, and ∛(8\times27) = ∛216 = 6. Same answer. Copy

Just like with simplifying square roots, 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.

2

Quotient Rule

Split quotients inside radicals

Example: ∛64 \div ∛8 = 4 \div 2 = 2, and ∛(64\div8) = ∛8 = 2. Copy

The mirror of the product rule: take the cube root of a fraction by rooting the numerator and denominator separately.

3

Power Rule

Cubing and cube root cancel each other

Example: ∛(5³) = ∛125 = 5. Cubing and cube-rooting undo each other. Copy

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) = -3.

4

Self-Cube Rule

Cubing a cube root returns the radicand

Example: (∛5)³ = 5. (∛12)³ = 12. It always works. Copy

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.

5

Identity Values

The cube roots of 0 and 1 never change

Example: No matter the context, ∛1 is always 1 and ∛0 is always 0. Copy

These are the fixed points of the cube root function. ∛1 = 1 because 1³ = 1, and ∛0 = 0 because 0³ = 0.

Working with higher degree root indices beyond 3? Compute 4th, 5th, and general nth roots online →

Methods

How to Calculate Racine Cubique: Different Methods

Four reliable ways to find cube roots:

Method 1

1. Using This Racine Cubique Calculator (Fastest)

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.

Method 2

2. Prime Factorization Method (Best for Perfect Cubes)

This method breaks a number into its prime factors and groups them in sets of three.

Example: Find ∛216

Step 1: Break 216 into prime factors: 216 = 2 × 2 × 2 × 3 × 3 × 3
Step 2: Group into sets of three: (2 × 2 × 2) × (3 × 3 × 3)
Step 3: Take one number from each group: 2 × 3 = 6
So ∛216 = 6.

Extended Example: Find ∛3375

Decompose into prime factors: 3375 = 3 × 3 × 3 × 5 × 5 × 5
Group into sets of three: (3 × 3 × 3) × (5 × 5 × 5) = 3³ × 5³
Take one factor from each group: 3 × 5 = 15
Therefore, ∛3375 = 15.

Method 3

4. Estimation Method (Fastest by Hand)

Place the number between two known cubes:

Example: Estimate ∛50

3³ = 27
4³ = 64
Since 50 is between 27 and 64, ∛50 is between 3 and 4, closer to 4.
The real answer is about 3.68, so this trick gets you close fast.

273³

50∛50 ≈ 3.68

644³

Perfect Cubes

Perfect Cubes: The Essential Memorization List

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.

1³ = 1

∛1 = 1

2³ = 8

∛8 = 2

3³ = 27

∛27 = 3

4³ = 64

∛64 = 4

5³ = 125

∛125 = 5

6³ = 216

∛216 = 6

7³ = 343

∛343 = 7

8³ = 512

∛512 = 8

9³ = 729

∛729 = 9

10³ = 1,000

∛1,000 = 10

11³ = 1,331

∛1,331 = 11

12³ = 1,728

∛1,728 = 12

👉 Scroll to browse table ↓

Number (n)	Cube (n³)	Cube Root (∛n³)
1	1	1
2	8	2
3	27	3
4	64	4
5	125	5
6	216	6
7	343	7
8	512	8
9	729	9
10	1,000	10
11	1,331	11
12	1,728	12
15	3,375	15
20	8,000	20
25	15,625	25
50	125,000	50
100	1,000,000	100

Negative Numbers

Cube Roots of Negative Numbers (Why They're Real, Unlike Square Roots)

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.

Square Root (Index 2)

\sqrt(-16) = No Real Root

Requires the imaginary unit i (±4i) because squaring any real number produces a non-negative result.

Cube Root (Index 3)

∛(-64) = -4

100% Real solution! Multiplying three negative numbers gives a negative product: (-4) × (-4) × (-4) = -64.

Example: ∛(−64)

∛(−64) = −4, because (−4) × (−4) × (−4) = −64

∛(-x) = -∛x

That's the full answer. No "i," no imaginary unit, nothing extra needed.

Key Difference from Square Roots

Square roots of negative numbers require the imaginary unit i, but cube roots handle negatives just fine on their own.

Cube Root of Decimals & Cube Root of Fractions

Cube Root of Decimals

To find the cube root of a decimal, convert it into a fraction or power of ten, simplify, and evaluate the root.

Example: ∛0.008

0.008 = 8 / 1000 = 8 × 10⁻³
∛(8 / 1000) = ∛8 / ∛1000 = 2 / 10
2 / 10 = 0.2

Cube Root of Fractions

Apply the quotient property by taking the cube root of the numerator and denominator separately: ∛(a/b) = ∛a / ∛b.

Example: ∛(8 / 27)

Apply quotient rule: ∛(8 / 27) = ∛8 / ∛27
Evaluate numerator: ∛8 = 2
Evaluate denominator: ∛27 = 3
Result: 2 / 3 ≈ 0.6667

Simplification

Cube Root Simplification Table

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 = 2³ × 2 = 8 × 2
∛24	2∛3	24 = 2³ × 3 = 8 × 3
∛32	2∛4	32 = 2³ × 4 = 8 × 4
∛40	2∛5	40 = 2³ × 5 = 8 × 5
∛48	2∛6	48 = 2³ × 6 = 8 × 6
∛54	3∛2	54 = 3³ × 2 = 27 × 2
∛56	2∛7	56 = 2³ × 7 = 8 × 7
∛72	2∛9	72 = 2³ × 9 = 8 × 9
∛108	3∛4	108 = 3³ × 4 = 27 × 4
∛128	4∛2	128 = 4³ × 2 = 64 × 2

The trick: find the largest perfect cube that divides evenly into your number, pull it out, and the rest stays under the radical.

Comparison

Difference Between Racine Cubique and Racine Carrée

A quick side-by-side comparison of 2D vs 3D radicals:

Property / Feature	Cube Root	Square Root
Symbol	∛ (Index 3)	√ (Index 2)
Exponent Form	x^(1/3)	x^(1/2)
Negative numbers	Real answer exists (∛-64 = -4)	No real answer (needs "i")
Multiplications	3 (x × x × x)	2 (x × x)
Number of Real Roots	Exactly 1 for all real numbers	1 for positive / 0 for negative
Graph Symmetry	Odd function (symmetric about origin)	Defined only for x ≥ 0
Output Sign	Always matches input sign (∛(-8) = -2)	Always non-negative (principal root)
Example	∛64 = 4 (4×4×4=64)	√64 = 8 (8×8=64)

Need to evaluate square roots instead? Calculate square roots with exact decimals and simplified radicals →

Reference Tables

Complete Reference Table: Cube Roots 1 to 100

Bookmark this section for quick lookups. Perfect cubes are highlighted in bold with a badge. Values are rounded to three decimal places.

👉 Scroll to see all 100 values ↓

Number (n)	Cube Root (∛n)	Simplified Radical	Type
1	1	`1`	Perfect Cube
2	1.2599	`∛2`	Irrational
3	1.4422	`∛3`	Irrational
4	1.5874	`∛4`	Irrational
5	1.7100	`∛5`	Irrational
6	1.8171	`∛6`	Irrational
7	1.9129	`∛7`	Irrational
8	2	`2`	Perfect Cube
9	2.0801	`∛9`	Irrational
10	2.1544	`∛10`	Irrational
11	2.2240	`∛11`	Irrational
12	2.2894	`∛12`	Irrational
13	2.3513	`∛13`	Irrational
14	2.4101	`∛14`	Irrational
15	2.4662	`∛15`	Irrational
16	2.5198	`2∛2`	Irrational
17	2.5713	`∛17`	Irrational
18	2.6207	`∛18`	Irrational
19	2.6684	`∛19`	Irrational
20	2.7144	`∛20`	Irrational
21	2.7589	`∛21`	Irrational
22	2.8020	`∛22`	Irrational
23	2.8439	`∛23`	Irrational
24	2.8845	`2∛3`	Irrational
25	2.9240	`∛25`	Irrational
26	2.9625	`∛26`	Irrational
27	3	`3`	Perfect Cube
28	3.0366	`∛28`	Irrational
29	3.0723	`∛29`	Irrational
30	3.1072	`∛30`	Irrational
31	3.1414	`∛31`	Irrational
32	3.1748	`2∛4`	Irrational
33	3.2075	`∛33`	Irrational
34	3.2396	`∛34`	Irrational
35	3.2711	`∛35`	Irrational
36	3.3019	`∛36`	Irrational
37	3.3322	`∛37`	Irrational
38	3.3620	`∛38`	Irrational
39	3.3912	`∛39`	Irrational
40	3.4200	`2∛5`	Irrational
41	3.4482	`∛41`	Irrational
42	3.4760	`∛42`	Irrational
43	3.5034	`∛43`	Irrational
44	3.5303	`∛44`	Irrational
45	3.5569	`∛45`	Irrational
46	3.5830	`∛46`	Irrational
47	3.6088	`∛47`	Irrational
48	3.6342	`2∛6`	Irrational
49	3.6593	`∛49`	Irrational
50	3.6840	`∛50`	Irrational
51	3.7084	`∛51`	Irrational
52	3.7325	`∛52`	Irrational
53	3.7563	`∛53`	Irrational
54	3.7798	`3∛2`	Irrational
55	3.8030	`∛55`	Irrational
56	3.8259	`2∛7`	Irrational
57	3.8485	`∛57`	Irrational
58	3.8709	`∛58`	Irrational
59	3.8930	`∛59`	Irrational
60	3.9149	`∛60`	Irrational
61	3.9365	`∛61`	Irrational
62	3.9579	`∛62`	Irrational
63	3.9791	`∛63`	Irrational
64	4	`4`	Perfect Cube
65	4.0207	`∛65`	Irrational
66	4.0412	`∛66`	Irrational
67	4.0615	`∛67`	Irrational
68	4.0817	`∛68`	Irrational
69	4.1016	`∛69`	Irrational
70	4.1213	`∛70`	Irrational
71	4.1408	`∛71`	Irrational
72	4.1602	`2∛9`	Irrational
73	4.1793	`∛73`	Irrational
74	4.1983	`∛74`	Irrational
75	4.2172	`∛75`	Irrational
76	4.2358	`∛76`	Irrational
77	4.2543	`∛77`	Irrational
78	4.2727	`∛78`	Irrational
79	4.2908	`∛79`	Irrational
80	4.3089	`2∛10`	Irrational
81	4.3267	`3∛3`	Irrational
82	4.3445	`∛82`	Irrational
83	4.3621	`∛83`	Irrational
84	4.3795	`∛84`	Irrational
85	4.3968	`∛85`	Irrational
86	4.4140	`∛86`	Irrational
87	4.4310	`∛87`	Irrational
88	4.4480	`2∛11`	Irrational
89	4.4647	`∛89`	Irrational
90	4.4814	`∛90`	Irrational
91	4.4979	`∛91`	Irrational
92	4.5144	`∛92`	Irrational
93	4.5307	`∛93`	Irrational
94	4.5468	`∛94`	Irrational
95	4.5629	`∛95`	Irrational
96	4.5789	`2∛12`	Irrational
97	4.5947	`∛97`	Irrational
98	4.6104	`∛98`	Irrational
99	4.6261	`∛99`	Irrational
100	4.6416	`∛100`	Irrational

Want to verify whether any specific integer is an exact cube? Test integers with our instant perfect cube checker →

Applications

Real-Life Applications of Racine Cubique

While square roots deal with 2D problems, cube roots shine whenever the problem involves three dimensions:

Architecture & 3D Design

Architects and designers use cube roots to find the edge length of cubic spaces from a known volume. For example, ∛64 = 4 m per side for a 64 m³ room.

Shipping & Packaging

Logistics companies calculate box dimensions from a target volume using cube roots to optimize parcel packaging and dimensional weight.

Physics & Engineering

Formulas in fluid dynamics, thermodynamics, and astronomy (such as planetary orbital periods and spherical radii) rely heavily on cube roots.

Finance & Growth Rates

Calculating 3-year compound annual growth rates (CAGR) requires taking the cube root of the total growth multiplier: r = ∛(End/Start) - 1.

Everyday Measuring & DIY

Finding side lengths from container capacities (e.g. aquarium volume, storage bins) without measuring tools.

Under the Hood

How Does This Cube Root Calculator Work?

1

Client-Side High Performance

All calculations execute directly in your browser with zero latency and 100% data privacy.

2

Exact Integer Detection

Instantly checks whether the radicand is a perfect cube and outputs exact integers without rounding drift.

3

Radical Simplification Engine

Performs prime factorization in real time to extract triple factors, rendering exact forms like ∛24 = 2∛3.

4

Full Real Negative Support

Applies the mathematical odd symmetry rule ∛(-x) = -∛x, returning exact real solutions for negative inputs.

5

Arbitrary Precision & Step Workings

Provides clean decimal approximations and comprehensive step-by-step mathematical reasoning.

FAQs

Frequently Asked Questions

What is a cube root?

A cube root of a number is a value that, when multiplied by itself three times, produces the original number. In mathematical notation, the cube root of x is written as ∛x or x^(1/3). For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27.

Can you take the cube root of a negative number?

Yes! Unlike square roots, cube roots of negative numbers are real numbers because multiplying three negative numbers results in a negative product. The rule is ∛(-x) = -∛x. For example, ∛(-8) = -2, because (-2) × (-2) × (-2) = -8.

What is the difference between cubing and taking a cube root?

Cubing and taking the cube root are inverse operations. Cubing a number means raising it to the power of 3 (for example, 4³ = 64). Taking the cube root reverses this process to recover the original base value: ∛64 = 4.

What is a perfect cube?

A perfect cube is an integer that results from multiplying an integer by itself three times. Examples include 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. The cube root of a perfect cube is always a whole number.

How do you calculate a cube root without a calculator?

For perfect cubes, use prime factorization: break the number into prime factors, group identical factors into sets of three, and extract one from each group (e.g. 3375 = 3³ × 5³ = 15). For non-perfect cubes, use estimation between the two nearest known perfect cubes.

What is the cube root symbol?

The cube root symbol is ∛ (Unicode character U+221B). It consists of a standard radical sign with a small index 3 in the upper-left notch. In software code it is written cbrt(x), and in LaTeX as \sqrt[3]{x}.

How do you type the cube root symbol?

On Windows, type 221B then press Alt+X in supported apps, or use Character Map. On Mac, press Cmd+Ctrl+Space to open Character Viewer and search 'cube root'. In HTML, use the entity ∛ or copy and paste ∛ directly.

What is ∛0?

The cube root of 0 is exactly 0, because 0 × 0 × 0 = 0.

Is the cube root of 2 rational or irrational?

The cube root of 2 (∛2 ≈ 1.2599) is an irrational number. It cannot be expressed as an exact fraction of two integers, and its decimal expansion continues infinitely without repeating.

What is the cube root of 1?

The real cube root of 1 is 1, since 1 × 1 × 1 = 1. (In complex mathematics, there are also two non-real complex cube roots of unity: e^(2πi/3) and e^(-2πi/3)).

Written & Verified by Syed Muhammad Ahmed Founder & Creator

Summary

A Quick Recap

The Bottom Line

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.