Square Root
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Complete Guide
What Is Racine Carrée (Square Root)?
Racine carrée is the French word for square root, and it's the term thousands of people type into Google every month — even when they're searching in English. If that's how you found this page, you're in the right place. Our racine carree calculator above gives you the answer in one click, and everything below explains what a square root actually is, how to work one out by hand, and why this little math operation shows up everywhere from geometry class to engineering.
A square root is a number that, when multiplied by itself, gives you another number. Here's the simplest way to picture it: if you multiply 5 by itself (5 × 5), you get 25. So we say the square root of 25 is 5. This same principle extends to higher-degree roots — for example, a cube root (racine cubique) multiplies a number by itself three times, and an nth root generalizes the idea to any degree.
In French, "racine carrée" means exactly this — square root. So when someone searches "racine carree calculator," they're looking for the same tool as someone who searches "square root calculator." Same math, different language.
√9 = 3 because 3 × 3 = 9
√16 = 4 because 4 × 4 = 16
√100 = 10 because 10 × 10 = 100
How To Use
How to Use the Racine Carree Calculator
Getting your answer takes three steps:
1
Type Your Number
Type your number into the input box above.
2
Click Calculate
Hit the Calculate button to process your number.
3
Read Your Result
The calculator shows the square root instantly, plus a simplified form when one exists.
That's the whole process. You don't need to know any formulas to use the tool — it does the work for you. The rest of this page is here for anyone who wants to actually understand how the answer is found, not just get the number.
Notation
Square Root Notation: What Does the √ Symbol Mean?
The symbol √ is called the radical sign, and it's how we write "square root" in math.
√9 "the square root of 9"
9 = radicand The number under the symbol
3 = root The answer you get
So when you see √49 = 7, you're reading: "the square root of 49 equals 7."
You'll sometimes also see a square root written as a fraction power:
x^(1/2) = √x
Both mean the exact same thing. Calculators and computer programs often use the exponent form because it's easier to type.
Formula
What Is the Formula for Square Root?
There isn't one single formula that spits out a square root the way there is for, say, the area of a circle. Instead, a square root is defined by this relationship:
If x = √y, then x × x = y (or x² = y)
In plain words: whatever number you get as the square root, multiplying it by itself has to bring you back to your original number. That relationship is what every method of calculating a square root — by hand, on a calculator, or with a computer — is built around.
Core Laws
The Five Core Laws of Square Roots
Once you know the basics, these five rules make working with square roots much faster. They also apply to cube roots and nth roots with slight modifications — master them here and you'll recognize the pattern everywhere.
1
Product Rule
Split products inside radicals
√(a × b) = √a × √b
Example: √4 × √9 = 2 × 3 = 6, and √(4×9) = √36 = 6. Same answer either way.
The product rule lets you break a complicated square root into simpler pieces. Instead of finding √36 directly, you can split 36 into 4 × 9, take each root separately, and multiply the results. This is the foundation of simplifying radicals — whenever you see a number under the radical sign, look for perfect square factors you can pull out. It's how our calculator simplifies expressions like √72 into 6√2.
2
Quotient Rule
Split quotients inside radicals
√(a ÷ b) = √a ÷ √b
Example: √16 ÷ √4 = 4 ÷ 2 = 2, and √(16÷4) = √4 = 2.
The quotient rule is the product rule's mirror image — it lets you take the square root of a fraction by rooting the top and bottom separately. This is particularly useful in algebra when you need to simplify expressions like √(50/2). Instead of dividing first, you can write √50 ÷ √2, simplify √50 to 5√2, and then cancel √2 to get 5.
3
Power Rule
Squaring and square root cancel each other
√(a²) = a (for a ≥ 0)
Example: √(7²) = √49 = 7. The square and the square root undo each other.
This is the most intuitive law: squaring a number and then taking its square root brings you right back to where you started. Think of them as mathematical inverses — one builds a square, the other finds the side length. This relationship is why the racine carrée calculator can "reverse" any squaring operation. Note that for negative inputs, √(a²) gives |a| (the absolute value), because a square root always returns the non-negative result.
4
Self-Multiplication Rule
A root times itself equals the radicand
√a × √a = a
Example: √7 × √7 = 7. √13 × √13 = 13. It always works.
This rule is really just the definition of a square root stated in reverse: if √a is "the number that multiplied by itself gives a," then multiplying √a by √a must give you a. It's especially handy for rationalizing denominators — when you have 1/√5, you multiply top and bottom by √5 to get √5/5, eliminating the radical from the denominator. You'll see this trick constantly in algebra and precalculus.
5
Identity Values
The square roots of 0 and 1 never change
√1 = 1 and √0 = 0
Example: No matter what context — algebra, physics, finance — √1 is always 1 and √0 is always 0.
These are the two "fixed point" values for the square root function. √1 = 1 because 1 × 1 = 1, and √0 = 0 because 0 × 0 = 0. While they look trivial, they're important in proofs and as boundary conditions. For instance, when checking whether a formula works at its extremes, plugging in 0 or 1 should always give a sensible answer — and these identities guarantee that it does.
Methods
How to Calculate Racine Carrée — Different Methods
There's more than one way to find a square root. Here are the four you're most likely to see in school or at work.
1. Using This Racine Carrée Calculator (Fastest)
Type the number in, click calculate, done. This is the method almost everyone reaches for once they know it, especially with decimals or large numbers. You can also use our cube root calculator for third-degree roots or the nth root calculator for any custom degree.
2. Prime Factorization Method (Best for Perfect Squares)
This method works by breaking a number down into its prime factors and pairing them up.
Example: Find √144
- Step 1 — Break 144 into prime factors: 144 = 2 × 2 × 2 × 2 × 3 × 3
- Step 2 — Pair up the matching factors: (2 × 2) × (2 × 2) × (3 × 3)
- Step 3 — Take one number from each pair: 2 × 2 × 3 = 12
- So √144 = 12.
3. Long Division Method (Best for Decimals and Large Numbers)
This is the pencil-and-paper method taught in most math classes for numbers that aren't perfect squares. It's a step-by-step process similar to regular long division, and it gives you a decimal answer to as many places as you want.
4. Estimation Method (Fastest by Hand, No Tools)
If you just need a rough answer, find the two perfect squares your number sits between.
Example: Estimate √20
- 4² = 16
- 5² = 25
- Since 20 is between 16 and 25, √20 is between 4 and 5 — closer to 4.5.
- The real answer is about 4.47, so this quick trick gets you very close.
Positive Numbers
Square Roots of Positive Numbers
Every positive number has a square root, and it's always positive (this is called the principal root, which is what our calculator shows).
Scroll horizontally →
| Number | Square Root | Why |
|---|---|---|
| 4 | 2 | 2 × 2 = 4 |
| 25 | 5 | 5 × 5 = 25 |
| 64 | 8 | 8 × 8 = 64 |
| 2 | ≈1.414 | Not a whole number, but still a real, positive answer |
Note that technically, every positive number has two square roots — a positive one and a negative one (for example, both 5 and -5 give you 25 when squared). But by convention, when we write √25, we mean the positive root, 5. That's what "principal root" means.
Imaginary Numbers
Square Roots of Negative Numbers and the Imaginary Unit
You can't take the square root of a negative number and get a regular (real) number back. Why? Because multiplying any positive number by itself gives a positive result, and multiplying any negative number by itself also gives a positive result (a negative times a negative is positive).
To handle this, mathematicians created the imaginary unit, written as i, where:
i = √(−1)
Using this, we can express the square root of any negative number.
Example: √(−81)
- √(−81) = √81 × √(−1) = 9 × i = 9i
So the square root of −81 is written as 9i — not a "real" number in the everyday sense, but a valid mathematical answer that's used constantly in advanced math, physics, and electrical engineering.
Comparison
Difference Between Racine Carrée and Racine Cubique
These two terms get mixed up a lot. Racine carrée (square root) asks: what number multiplied by itself gives me this value? Racine cubique (cube root) asks: what number multiplied by itself three times gives me this value?
Scroll horizontally →
| Square Root | Cube Root | |
|---|---|---|
| Symbol | √ | ∛ |
| Multiplications | 2 (x × x) | 3 (x × x × x) |
| Example | √25 = 5 | ∛27 = 3 |
Applications
Real-Life Applications of Racine Carrée
Square roots are foundational in modern life. Here are five areas where understanding them gives you a real advantage:
Construction & Architecture
Used in the Pythagorean theorem (√(a² + b²)) to check if corners are perfectly square and to measure structural rafters.
Finance & Investing
Standard deviation—the key metric for measuring portfolio risk—is the square root of variance.
Physics & Engineering
Found in critical formulas for velocity, kinetic energy, and wave speed across all STEM disciplines.
Computer Science
Essential for calculating Euclidean distances in game physics, machine learning, and image processing.
Everyday Measuring
Used to find side lengths when you already know the area of a square lot, carpet, or garden patch.
Simplification
Square Root Simplification Table
Not every square root is a perfect square, but many can still be simplified into a smaller, cleaner form by pulling out any perfect-square factors.
Scroll horizontally →
| Expression | Simplified Form | Why |
|---|---|---|
| √8 | 2√2 | 8 = 4 × 2, and √4 = 2 |
| √12 | 2√3 | 12 = 4 × 3 |
| √18 | 3√2 | 18 = 9 × 2 |
| √20 | 2√5 | 20 = 4 × 5 |
| √50 | 5√2 | 50 = 25 × 2 |
| √72 | 6√2 | 72 = 36 × 2 |
| √98 | 7√2 | 98 = 49 × 2 |
The trick is always the same: look for the largest perfect square that divides evenly into your number, pull it out, and the rest stays under the radical sign.
Reference Tables
Complete Reference Table — Square Roots 1 to 100
Bookmark this section for quick lookups. Perfect squares are highlighted in bold. Values are rounded to three decimal places. For cube root reference tables, see our dedicated racine cubique page.
Scroll to see all 100 values ↓
| Number (n) | Square Root (√n) | Type |
|---|---|---|
| 1 | 1.000 | Perfect Square |
| 2 | 1.414 | Irrational |
| 3 | 1.732 | Irrational |
| 4 | 2.000 | Perfect Square |
| 5 | 2.236 | Irrational |
| 6 | 2.449 | Irrational |
| 7 | 2.646 | Irrational |
| 8 | 2.828 | Irrational |
| 9 | 3.000 | Perfect Square |
| 10 | 3.162 | Irrational |
| 11 | 3.317 | Irrational |
| 12 | 3.464 | Irrational |
| 13 | 3.606 | Irrational |
| 14 | 3.742 | Irrational |
| 15 | 3.873 | Irrational |
| 16 | 4.000 | Perfect Square |
| 17 | 4.123 | Irrational |
| 18 | 4.243 | Irrational |
| 19 | 4.359 | Irrational |
| 20 | 4.472 | Irrational |
| 21 | 4.583 | Irrational |
| 22 | 4.690 | Irrational |
| 23 | 4.796 | Irrational |
| 24 | 4.899 | Irrational |
| 25 | 5.000 | Perfect Square |
| 26 | 5.099 | Irrational |
| 27 | 5.196 | Irrational |
| 28 | 5.292 | Irrational |
| 29 | 5.385 | Irrational |
| 30 | 5.477 | Irrational |
| 31 | 5.568 | Irrational |
| 32 | 5.657 | Irrational |
| 33 | 5.745 | Irrational |
| 34 | 5.831 | Irrational |
| 35 | 5.916 | Irrational |
| 36 | 6.000 | Perfect Square |
| 37 | 6.083 | Irrational |
| 38 | 6.164 | Irrational |
| 39 | 6.245 | Irrational |
| 40 | 6.325 | Irrational |
| 41 | 6.403 | Irrational |
| 42 | 6.481 | Irrational |
| 43 | 6.557 | Irrational |
| 44 | 6.633 | Irrational |
| 45 | 6.708 | Irrational |
| 46 | 6.782 | Irrational |
| 47 | 6.856 | Irrational |
| 48 | 6.928 | Irrational |
| 49 | 7.000 | Perfect Square |
| 50 | 7.071 | Irrational |
| 51 | 7.141 | Irrational |
| 52 | 7.211 | Irrational |
| 53 | 7.280 | Irrational |
| 54 | 7.348 | Irrational |
| 55 | 7.416 | Irrational |
| 56 | 7.483 | Irrational |
| 57 | 7.550 | Irrational |
| 58 | 7.616 | Irrational |
| 59 | 7.681 | Irrational |
| 60 | 7.746 | Irrational |
| 61 | 7.810 | Irrational |
| 62 | 7.874 | Irrational |
| 63 | 7.937 | Irrational |
| 64 | 8.000 | Perfect Square |
| 65 | 8.062 | Irrational |
| 66 | 8.124 | Irrational |
| 67 | 8.185 | Irrational |
| 68 | 8.246 | Irrational |
| 69 | 8.307 | Irrational |
| 70 | 8.367 | Irrational |
| 71 | 8.426 | Irrational |
| 72 | 8.485 | Irrational |
| 73 | 8.544 | Irrational |
| 74 | 8.602 | Irrational |
| 75 | 8.660 | Irrational |
| 76 | 8.718 | Irrational |
| 77 | 8.775 | Irrational |
| 78 | 8.832 | Irrational |
| 79 | 8.888 | Irrational |
| 80 | 8.944 | Irrational |
| 81 | 9.000 | Perfect Square |
| 82 | 9.055 | Irrational |
| 83 | 9.110 | Irrational |
| 84 | 9.165 | Irrational |
| 85 | 9.220 | Irrational |
| 86 | 9.274 | Irrational |
| 87 | 9.327 | Irrational |
| 88 | 9.381 | Irrational |
| 89 | 9.434 | Irrational |
| 90 | 9.487 | Irrational |
| 91 | 9.539 | Irrational |
| 92 | 9.592 | Irrational |
| 93 | 9.644 | Irrational |
| 94 | 9.695 | Irrational |
| 95 | 9.747 | Irrational |
| 96 | 9.798 | Irrational |
| 97 | 9.849 | Irrational |
| 98 | 9.899 | Irrational |
| 99 | 9.950 | Irrational |
| 100 | 10.000 | Perfect Square |
Perfect Squares
Perfect Squares — The Essential Memorization List
A perfect square is a number you get by multiplying a whole number by itself. Knowing these by heart makes square roots — and a lot of algebra — much faster.
| Number | Square | Number | Square |
|---|---|---|---|
| 1 | 1 | 11 | 121 |
| 2 | 4 | 12 | 144 |
| 3 | 9 | 13 | 169 |
| 4 | 16 | 14 | 196 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 16 | 256 |
| 7 | 49 | 17 | 289 |
| 8 | 64 | 18 | 324 |
| 9 | 81 | 19 | 361 |
| 10 | 100 | 20 | 400 |
If you can recognize these on sight, spotting perfect squares (and simplifying square roots) becomes second nature.
FAQs
Frequently Asked Questions
How do I use the Racine Carree Calculator?
Type any number into the box above, click Calculate, and the tool instantly shows you the square root — including a simplified form when one exists and support for negative numbers.
Can I calculate square roots manually?
Yes. For perfect squares, prime factorization works well. For everything else, the long division method or the estimation trick (finding which two perfect squares your number sits between) will get you there — just more slowly than a calculator.
What does "racine carrée" mean?
It's French for "square root." A racine carrée calculator and a square root calculator do exactly the same job — the name just reflects that a lot of English-speaking users search using the French phrase.
Why is the keyword "racine carrée" in French?
It's a quirk of how people search. Even though this site and its explanations are written in English, "racine carrée" shows up as a common search term among US, UK, and Canadian users — likely from French class, Canadian bilingual schooling, or simply how the term was first learned.
Can I calculate square roots of decimals or large numbers?
Yes. The calculator handles decimals, whole numbers, and large values with high precision, so you don't need to worry about the size or format of what you're entering.
Can the calculator handle negative numbers?
Yes. When you enter a negative number, the calculator returns the result using the imaginary unit "i," as explained in the section above.
Is this calculator free to use?
Yes, completely — no sign-up, no downloads, no limits on how many times you use it.
Summary
A Quick Recap
The Bottom Line
A square root answers one simple question: what number, multiplied by itself, gives you this value? Once that idea clicks, everything else on this page — the notation, the rules, the manual methods, even the imaginary numbers — is just building on that one concept. Use the calculator above whenever you need a fast, accurate answer, and come back to this guide anytime you want to double-check your understanding or work a problem out by hand.