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Racine Carrée Calculator (Square Root Calculator)

Calculate square roots instantly, simplify radicals, find exact and decimal answers, and learn square root concepts step by step.

Square Root

Enter any number to get its exact value, simplified radical, and decimal approximation.

√
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Square & Square Root Reference (1–100)

Perfect squares are highlighted. Scroll to see all 100 rows.

NumberSquare (n²)Square Root (√n)
111
241.4142
391.7321
4162
5252.2361
6362.4495
7492.6458
8642.8284
9813
101003.1623
111213.3166
121443.4641
131693.6056
141963.7417
152253.8730
162564
172894.1231
183244.2426
193614.3589
204004.4721
214414.5826
224844.6904
235294.7958
245764.8990
256255
266765.0990
277295.1962
287845.2915
298415.3852
309005.4772
319615.5678
321,0245.6569
331,0895.7446
341,1565.8310
351,2255.9161
361,2966
371,3696.0828
381,4446.1644
391,5216.2450
401,6006.3246
411,6816.4031
421,7646.4807
431,8496.5574
441,9366.6332
452,0256.7082
462,1166.7823
472,2096.8557
482,3046.9282
492,4017
502,5007.0711
512,6017.1414
522,7047.2111
532,8097.2801
542,9167.3485
553,0257.4162
563,1367.4833
573,2497.5498
583,3647.6158
593,4817.6811
603,6007.7460
613,7217.8102
623,8447.8740
633,9697.9373
644,0968
654,2258.0623
664,3568.1240
674,4898.1854
684,6248.2462
694,7618.3066
704,9008.3666
715,0418.4261
725,1848.4853
735,3298.5440
745,4768.6023
755,6258.6603
765,7768.7178
775,9298.7750
786,0848.8318
796,2418.8882
806,4008.9443
816,5619
826,7249.0554
836,8899.1104
847,0569.1652
857,2259.2195
867,3969.2736
877,5699.3274
887,7449.3808
897,9219.4340
908,1009.4868
918,2819.5394
928,4649.5917
938,6499.6437
948,8369.6954
959,0259.7468
969,2169.7980
979,4099.8489
989,6049.8995
999,8019.9499
10010,00010

Worked Examples

🟦

√144 = 12

A perfect square — 12 × 12 = 144. Useful for finding the side of a square with area 144.

🧮

√72 = 6√2 ≈ 8.49

Not a perfect square, but it simplifies cleanly because 72 = 36 × 2.

♾️

√2 ≈ 1.41421356

The classic irrational number — its decimal never ends and never repeats.

💾

√1024 = 32

A perfect square that shows up constantly in computing (1024 = 2¹⁰).

Multilingual Terminology

Racine carrée is the French term for square root. You will also see it called raíz cuadrada (Spanish), Quadratwurzel (German), radice quadrata (Italian), and raiz quadrada (Portuguese). In every language the symbol is the same radical sign, √, and the math is identical: the value that multiplies by itself to give the number under the root.

What Is a Racine Carrée (Square Root)?

Racine carrée is simply the French phrase for square root. The math behind both words is identical: the square root of a number x, written √x, is the value that gives x back when it is multiplied by itself. Because 12 × 12 = 144, the racine carrée of 144 is 12. Because 1.414… × 1.414… ≈ 2, the racine carrée of 2 is about 1.41421356. Whether your textbook says “racine carrée” or “square root,” you are asking the same question — what number, squared, lands on this value?

This page is a complete square-root workspace, not just a one-line answer box. It computes exact and decimal roots, simplifies radicals into forms like 6√2, checks whether a number is a perfect square, and shows the working at every step. It pairs naturally with the scientific calculator, the exponent calculator, and the cube root & nth-root calculator.

How to Calculate a Square Root

1 · Identify the radicand

The number sitting under the radical sign is the radicand. Type any value — a whole number, a decimal, or even a negative — and the calculator reads it instantly.

2 · Test for a perfect square

First the tool checks whether the radicand is a perfect square such as 1, 4, 9, 16, 25, 36, 49. If it is, the root is a clean whole number and no decimal is needed.

3 · Factor out perfect squares

If it is not a perfect square, the calculator finds the largest square factor and writes the answer in simplified radical form — √72 becomes 6√2 because 72 = 36 × 2.

4 · Approximate and verify

Finally it returns a decimal to your chosen precision (2 to 15 places) and squares the answer back so you can confirm the result matches the original number.

6 Ways to Use This Calculator

1

Find a clean root

Enter a perfect square — 144, 225, 625, 1024 — and read the exact whole-number answer with a squared verification.

2

Simplify a radical

Switch to Simplify Radical and turn √72, √50, or √288 into 6√2, 5√2, and 12√2 — complete with prime-factor working.

3

Check a perfect square

Use Perfect Square mode to get an instant Yes/No for any whole number, plus the two perfect squares that surround it.

4

Control decimal precision

Choose 2, 4, 6, 8, 10, or 15 decimal places — handy for everything from quick estimates to engineering-grade accuracy.

5

Explore with what-if

Tap the what-if chips to instantly test nearby values, the negative, or a ten-times-larger number without retyping.

6

Learn the method

Every answer ships with a step-by-step solution and number insights, so the tool teaches the technique, not just the result.

Perfect Squares & Simplifying Radicals

Know the perfect squares by heart. A perfect square is any integer that equals a whole number times itself — 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. When you can recognise these instantly, simplifying radicals becomes almost effortless because you can spot the square factor hidden inside the radicand.

Pull out the biggest square factor. To simplify √200, find the largest perfect square that divides 200 — that is 100 — and write √200 = √100 · √2 = 10√2. Using a smaller factor like 4 still works but leaves more cleanup: √200 = 2√50 = 2 · 5√2 = 10√2. Going straight for the largest square saves steps.

Keep exact and decimal forms side by side. The simplified radical 6√2 is exact and perfect for algebra; the decimal ≈ 8.485 is what you measure with. Real work often needs both, which is why this calculator always reports them together — much like the fraction calculator shows both fraction and decimal answers.

Applications of Square Roots

Geometry & distance

The Pythagorean theorem ends in a square root every time you solve for a side: c = √(a² + b²). Diagonals, screen sizes, and shortest paths all rely on it.

Statistics & data

Standard deviation is the square root of variance, and the margin of error shrinks with √n. Square roots quietly underpin nearly every measure of spread.

Physics & science

A pendulum's period is T = 2π√(L/g); orbital speed, RMS voltage, and free-fall time all invert squared quantities back to linear ones with a square root.

Construction & finance

Builders square up foundations using the 3-4-5 rule (√(3² + 4²) = 5); analysts scale portfolio risk with √(time). The same operation spans trades and markets.

Exact vs Decimal & Irrational Roots

Exact answers

When a root simplifies to a whole number or a clean radical (12, 6√2), that is an exact answer — no rounding, no lost precision. Keep exact forms through algebra.

Decimal approximations

Irrational roots like √2 ≈ 1.41421356 can only be written exactly as a radical; their decimals run forever. The calculator rounds to the precision you pick.

Why √2 is irrational

No fraction of two integers ever squares to exactly 2 — a fact the ancient Greeks proved by contradiction. So √2's decimal never terminates and never repeats.

Square roots of zero & negatives

√0 = 0 exactly. A negative radicand has no real root: √(−4) = 2i, an imaginary number built on i = √(−1). The calculator reports the imaginary form rather than dropping the sign.

Square Root Rules & Core Formulas

Definition

√x = y ⇔ y² = x (x ≥ 0)

Square the root and you get the radicand back — the rule every other identity rests on.

Product rule

√(a · b) = √a · √b

Split a radical across a product. This is exactly how radicals get simplified.

Quotient rule

√(a / b) = √a / √b

Distribute the root over a fraction so the top and bottom simplify separately.

Exponent form

√x = x^(1/2)

A square root is just a one-half power — useful when applying exponent laws.

Simplified form

√(k² · m) = k√m

Pull the largest perfect-square factor out front to reach simplest radical form.

Imaginary unit

√(−1) = i

Defines i so the root of any negative number can be written as √x · i.

Common Square Root Mistakes

Splitting a sum

√(a + b) is not √a + √b. Check it: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. The root only distributes over products, never sums.

Ignoring the negative root

When you solve x² = 25, both x = 5 and x = −5 work. The √ symbol returns only the positive (principal) root, so remember to add ± when solving equations.

Forcing a real answer from a negative

√(−9) is not −3 and not 3 — it is 3i. Reporting a real number for a negative radicand is a genuine error, not just a rounding choice.

Stopping at a non-simplest radical

Writing √72 as 2√18 is correct but unfinished, because 18 still hides a square (9). The simplest form is 6√2. Always factor until the radicand is square-free.

“Racine Carrée” Around the World

The square root goes by many names — racine carrée in French, raíz cuadrada in Spanish, Quadratwurzel in German, radice quadrata in Italian — but the radical sign √ and the underlying math are universal. This calculator is built for students, teachers, engineers, and anyone working with roots in any language. Every result is verified by squaring the answer, and every decimal is a faithful approximation of the exact value, so you can trust the numbers whether you call it a square root or a racine carrée.

Racine Carrée Calculator FAQs

"Racine carrée" is the French term for "square root." "Racine" means root and "carrée" means square, so the phrase translates literally as "square root." It refers to exactly the same operation in every language — finding the number that, multiplied by itself, gives the value under the radical sign √. You will also see it called raíz cuadrada (Spanish), Quadratwurzel (German), and radice quadrata (Italian), but the symbol √ and the math are identical worldwide.

The square root of a number x, written √x, is the value that produces x when multiplied by itself. For example, √144 = 12 because 12 × 12 = 144, and √2 ≈ 1.41421356 because 1.41421356 × 1.41421356 ≈ 2. The square root is the inverse of squaring a number. By convention the √ symbol returns the positive (principal) root, even though a positive number technically has both a positive and a negative square root.

First check whether the number is a perfect square (1, 4, 9, 16, 25, …) — if so, the root is a clean whole number. If not, factor out the largest perfect-square divisor to get a simplified radical such as √72 = 6√2, then read the decimal approximation. This calculator does all three steps automatically: it detects perfect squares, simplifies the radical, and returns a decimal to your chosen precision, with a squared verification line so you can confirm the answer.

A perfect square is a whole number that equals an integer multiplied by itself. The first perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 — that is, 1² through 12². The square root of a perfect square is always an exact whole number. Use the Perfect Square mode in this calculator to test any number instantly and see the two perfect squares that surround it.

Not within the real numbers. Any real number squared — positive or negative — is non-negative, so no real number squares to a negative value. The square root of a negative number is therefore imaginary: √(−1) is defined as i, and √(−4) = 2i. The calculator returns this imaginary form whenever you enter a negative radicand instead of incorrectly dropping the minus sign.

√2 is irrational because no fraction of two whole numbers ever squares to exactly 2 — a fact the ancient Greeks proved by contradiction. Since it cannot be written as a ratio of integers, its decimal expansion (1.41421356…) never terminates and never repeats. The same is true of √3, √5, √7, and the square root of any number that is not a perfect square; only an exact radical form captures the value precisely.

To simplify a square root, find the largest perfect-square factor of the radicand and pull its root out front. For √72, note that 72 = 36 × 2, so √72 = √36 · √2 = 6√2. For √200, 200 = 100 × 2, so √200 = 10√2. Keep factoring until the number left under the radical is square-free (has no perfect-square factor other than 1). The Simplify Radical mode shows this with full prime-factor working.

The square root of 144 is 12, because 12 × 12 = 144. 144 is a perfect square (it is 12²), so its racine carrée is an exact whole number with no decimal needed. Related perfect squares include √121 = 11, √169 = 13, and √100 = 10.

Yes. Switch to the Simplify Radical mode, enter a whole number, and the calculator reduces its square root to simplest radical form — for example √288 = 12√2 — and shows the prime factorisation and step-by-step working behind it. The default Square Root mode also displays the simplified radical alongside the exact and decimal answers for any non-negative integer.

An exact answer is the precise value with no rounding — either a whole number (√144 = 12) or a simplified radical (√72 = 6√2). A decimal answer is an approximation, because irrational roots like √2 = 1.41421356… run on forever and must be cut off at some number of places. Exact forms are best for algebra where precision must be preserved; decimal forms are best for measurement and real-world use. This calculator gives you both, and lets you set the decimal precision from 2 to 15 places.

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