The square root of a number m is the non-negative number whose square is m. The square root of 144 is 12, because 12 x 12 = 144. A number that is not a perfect square has an irrational root: the square root of 72 is 6 sqrt(2) in exact form, or about 8.48528137.
Principal square root
Calculations run in your browser. Inputs are not sent to our servers; anything you Save stays in this browser only.
Saved results (0)
How to Use the Square Root Calculator
- Type the number. Whole numbers, decimals and fractions all work. Write a fraction with a slash, such as 3/4.
- Pick the decimal places. Eight places suit most homework. Perfect squares show their exact root at any setting.
- Read both forms. The large figure is the decimal root. The exact simplified form is the answer most algebra courses want.
| Result | What it tells you |
|---|---|
| Principal square root | The non-negative root as a decimal. For a negative input it is the imaginary root, marked with i. |
| Exact simplified form | The radical with every perfect square factor taken out and no radical in a denominator, such as 6 sqrt(2) or sqrt(3) / 2. |
| Both square roots | The positive and the negative root. Both give your number when squared. |
| Perfect square? | Yes when the root is a whole number or an exact fraction. |
| Nearest perfect squares | The two consecutive whole-number squares your number lies between. |
| Prime factorization | For whole numbers, the prime factors the exact form is built from. |
| Check: root squared | The root multiplied by itself. It returns your number, give or take the last rounded digit. |
What Is a Square Root?
A square root of a number m is a number n whose square is m: n2 = m. Because 152 = 225, the number 15 is a square root of 225.
Every positive number has two square roots, one positive and one negative. (-15)2 is also 225. The radical sign means the positive one, called the principal square root. OpenStax Elementary Algebra states it this way: the square root of m is the non-negative number whose square is m. Zero has only one square root, 0.
A perfect square is the square of a whole number, such as 1, 4, 9, 16 and 25. Its root is a whole number. The square root of any other whole number is not a whole number, and its decimal never ends or repeats.
A square root is not half of a number. The square root of 16 is 4, not 8.
How Does the Square Root Calculator Work?
The calculator finds the decimal root, then builds the exact form with two properties of square roots.
- Read the input. A decimal or fraction is turned into a fraction in lowest terms. 12.5 becomes 25 / 2.
- Factor. The tool finds the prime factors. Each pair of equal primes is a perfect square factor and moves outside the radical. 72 = 23 x 32, so 2 x 3 = 6 comes out and one 2 stays in.
- Clear the denominator. For a fraction a / b, the tool uses sqrt(a / b) = sqrt(ab) / b. That leaves no radical in the denominator.
- Decimal value. The browser math library returns the root to about 15 significant digits. Perfect squares use the exact value.
- Negative inputs. With the imaginary unit i, where i2 = -1, the square root of -b is sqrt(b) i.
Square Root Examples: 72, 3/4 and -9
Square Root of 72
72 lies between 82 = 64 and 92 = 81, so the root is between 8 and 9. The largest perfect square factor of 72 is 36. sqrt(72) = sqrt(36) x sqrt(2) = 6 sqrt(2). As a decimal that is 8.48528137.
Square Root of 3/4
The quotient property splits the fraction: sqrt(3 / 4) = sqrt(3) / sqrt(4) = sqrt(3) / 2, or about 0.8660254. The denominator is a perfect square, so no radical is left below the line.
Square Root of 12.5
12.5 is 25 / 2. sqrt(25 / 2) = 5 / sqrt(2). Multiplying top and bottom by sqrt(2) gives 5 sqrt(2) / 2, or 3.53553391.
Square Root of -9
No real number squared gives -9. In the complex numbers, sqrt(-9) = sqrt(9) x i = 3i. The two roots are 3i and -3i.
Perfect Squares from 1 to 25
| n | n squared | n | n squared | n | n squared | n | n squared | n | n squared |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 4 | 3 | 9 | 4 | 16 | 5 | 25 |
| 6 | 36 | 7 | 49 | 8 | 64 | 9 | 81 | 10 | 100 |
| 11 | 121 | 12 | 144 | 13 | 169 | 14 | 196 | 15 | 225 |
| 16 | 256 | 17 | 289 | 18 | 324 | 19 | 361 | 20 | 400 |
| 21 | 441 | 22 | 484 | 23 | 529 | 24 | 576 | 25 | 625 |
Factors That Change a Square Root Result
The sign, the size and the form of the input decide what the answer looks like.
Sign of the Number
A positive number has two real roots. Zero has one. A negative number has none in the real numbers, and two imaginary roots.
Perfect Square Factors
A whole number with a perfect square factor simplifies. 200 = 100 x 2, so sqrt(200) = 10 sqrt(2). 31 has none, so sqrt(31) is already simplest. 32 = 16 x 2 gives 4 sqrt(2).
Scale by Powers of 100
Multiplying a number by 100 multiplies its root by 10. sqrt(20) is 4.47213595 and sqrt(2000) is 44.72135955. The same rule gives sqrt(0.01) = 0.1 and sqrt(0.0001) = 0.01.
Fractions and Decimals
A fraction is a perfect square only when its top and bottom are both perfect squares in lowest terms. 9 / 16 gives 3 / 4, and 1.44 gives 6 / 5 = 1.2.
Decimal Places
An irrational root must be cut off. 1.4142 squared is 1.99996164, not 2. More places bring the check closer.
Square Root Calculator vs Root Calculator and Cube Root Calculator
This page handles index 2 only and goes deeper on it. Two sister tools cover the other indexes, and there is some overlap between the three.
| Feature | This square root calculator | Root calculator | Cube root calculator |
|---|---|---|---|
| Index | 2 only | Any whole index | 3 only |
| Fraction input such as 3/4 | Yes, with exact form | No | No |
| Radical cleared from the denominator | Yes | No | No |
| Negative input | Imaginary roots in terms of i | Real root for odd indexes only | Real negative root |
| Both signed roots | Yes | Yes, for even indexes | Not applicable |
Use the root calculator for fourth, fifth or higher roots, and the cube root calculator for index 3. All three give the same decimal for a square root of a whole number.
When to Use a Square Root
Side of a Square from Its Area
A square with an area of 625 square feet has sides of sqrt(625) = 25 feet.
Right Triangles
The longest side of a right triangle is the square root of the sum of the squares of the other two. Sides 3 and 4 give sqrt(9 + 16) = 5. The Pythagorean theorem calculator does the full problem.
Solving X Squared Equals a Number
x2 = 72 has two solutions, x = 6 sqrt(2) and x = -6 sqrt(2). The quadratic equation calculator handles the general case.
Statistics
A standard deviation is the square root of a variance. A variance of 169 gives a standard deviation of 13.
Common Mistakes with Square Roots
1. Halving Instead of Taking the Root
sqrt(16) is 4, because 4 x 4 = 16. Half of 16 is 8, a different operation.
2. Forgetting the Negative Root
The radical sign gives the positive root only. An equation such as x2 = 225 has two answers, 15 and -15.
3. Splitting a Sum
sqrt(9 + 16) is sqrt(25) = 5. It is not sqrt(9) + sqrt(16) = 7. The product property works for multiplication, not addition.
4. Stopping the Simplification Early
sqrt(72) = sqrt(4) x sqrt(18) = 2 sqrt(18) still hides a factor of 9. The simplest form is 6 sqrt(2).
5. Leaving a Radical in the Denominator
5 / sqrt(2) is correct but not simplified. Multiply top and bottom by sqrt(2) to get 5 sqrt(2) / 2.
6. Calling the Root of a Negative an Error
sqrt(-9) is not a real number, but it is 3i in the complex numbers. Many calculators show an error because they work in real numbers only.
Accuracy and Limitations
The decimal root is accurate to about 15 to 16 significant digits, the limit of double-precision numbers in a browser. The exact form is exact.
What it calculates accurately
- The principal square root of any positive number, and of zero.
- The exact simplified radical for whole numbers, fractions and decimals with up to 6 decimal places, while numerator times denominator stays under one trillion.
- Exact perfect square detection for those inputs.
- Imaginary roots of negative numbers in the form b i.
- Babylonian steps and a squared check for every non-perfect square.
What it does not account for
- Variables or expressions such as sqrt(50x2).
- Square roots of complex numbers such as 3 + 4i.
- Indexes other than 2. Use the root or cube root page for those.
- Rounding rules for a specific textbook; use the rounding calculator to match one.
How We Calculate the Square Root
Frequently Asked Questions About Square Roots
What is the square root of 2?
It is about 1.41421356. The value is irrational, so the decimal never ends or repeats.
How do I simplify a square root?
Find the largest perfect square factor and take its root outside the radical. For 50, that factor is 25, so sqrt(50) = 5 sqrt(2), about 7.07106781.
Can a square root be negative?
The radical sign always means the non-negative root. Every positive number also has a negative square root, written with a minus sign in front of the radical.
What is the square root of a negative number?
It is not a real number. Using the imaginary unit i, the square root of -25 is 5i and the square root of -12 is 2 sqrt(3) i.
How do I find a square root without a calculator?
Guess, divide, and average. For 2, start at 2: the average of 2 and 2 / 2 is 1.5, the next step gives 1.416666667, and the third gives 1.414215686.
How do I estimate the square root of 60?
Find the perfect squares around it. 60 lies between 49 and 64, so its root lies between 7 and 8. The value is 7.74596669.
What is the square root of a fraction?
Take the root of the top and the bottom separately. sqrt(21 / 64) = sqrt(21) / 8, and sqrt(9 / 16) = 3 / 4.
Is the square root of 10 rational?
No. 10 is not a perfect square, so its square root, about 3.16227766, cannot be written as a fraction of whole numbers.
What is the square root of 0?
It is 0, because 0 x 0 = 0. Zero is the only number with exactly one square root.
Is my number saved anywhere?
No. The math runs in your browser. The Save button keeps a result only in your own browser storage.
Sources
- Square roots, the principal root and estimating between perfect squares (OpenStax Elementary Algebra 2e, section 9.1).
- Product and quotient properties and the simplified square root (OpenStax Elementary Algebra 2e, section 9.2).
- Rationalizing a denominator (OpenStax Elementary Algebra 2e, section 9.5).
- The imaginary unit and square roots of negative numbers (OpenStax Intermediate Algebra 2e, section 8.8).
- Newton method for approximating a square root (OpenStax Calculus Volume 1, section 4.9).
Related Calculators
Need another roots, powers or algebra tool?
Explore all math calculatorsThis calculator is for study and general math work. Decimal square roots of numbers that are not perfect squares are rounded values. For engineering or safety-critical work, confirm the result with the method your field requires. Spotted an error? Let us know.
Author
Shakeel Muzaffar is the Founder and Editor-in-Chief of MultiCalculators.com, bringing over 15 years of experience in digital publishing, product strategy, and online tool development. He leads the platform's editorial vision, ensuring every calculator meets strict standards for accuracy, usability, and real-world value. Shakeel personally oversees content quality, formula verification workflows, and the platform's commitment to publishing tools that are genuinely useful for students, professionals, and everyday users worldwide.
