Square Root Calculator
Square Root Calculator
Find the square root of any number in simplest radical form and in decimals to up to 15 places, or switch to a cube root or any nth root. Negative numbers give imaginary roots, and the working shows how the root is simplified and calculated.
The root curve
Between two whole number roots
Decimals are worked out with whole number arithmetic on as many digits as needed, then rounded, so every digit shown is correct. The simplest radical form takes the largest perfect square, cube or nth power out from under the root sign: 72 = 36 x 2, so β72 = 6β2. For decimals, the number is first written as a fraction. Even roots of negative numbers are imaginary and shown with i; odd roots of negative numbers are negative. The CSV, image and printed report contain the values you entered, nothing preloaded.
How to use this tool
Type a number and the square root appears at once, in simplest radical form where there is one and as a decimal. Change the root to a cube root or any root up to the 10th, and choose up to 15 decimal places. Negative numbers and decimals work too.
The cards give the rounded values, the reciprocal and the root as a power, and say whether the number is a perfect square. Below, the working shows Newton’s method converging, the curve and number lines show where the number sits, and the table lists the roots of the numbers around it.
How square roots are worked out
The square root of a number is the value that gives the number when multiplied by itself. Perfect squares such as 64 have whole number roots; every other whole number has an irrational root, a decimal that never ends or repeats, so it is written exactly with the root sign and simplified by taking out the largest perfect square factor.
For the decimal, the calculator scales the number up by a power of ten, finds the whole number root of that with exact arithmetic, and places the decimal point, so every digit shown is correct, not just approximate. The working shows Newton’s method instead, because it is how you would do it by hand: each pass averages the guess with the number divided by the guess.
Square roots and cube roots from 1 to 30
| Number | Square root | Simplest form | Cube root |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 1.4142 | β2 | 1.2599 |
| 3 | 1.7321 | β3 | 1.4422 |
| 4 | 2 | 2 | 1.5874 |
| 5 | 2.2361 | β5 | 1.7100 |
| 6 | 2.4495 | β6 | 1.8171 |
| 7 | 2.6458 | β7 | 1.9129 |
| 8 | 2.8284 | 2β2 | 2 |
| 9 | 3 | 3 | 2.0801 |
| 10 | 3.1623 | β10 | 2.1544 |
| 11 | 3.3166 | β11 | 2.2240 |
| 12 | 3.4641 | 2β3 | 2.2894 |
| 13 | 3.6056 | β13 | 2.3513 |
| 14 | 3.7417 | β14 | 2.4101 |
| 15 | 3.8730 | β15 | 2.4662 |
| 16 | 4 | 4 | 2.5198 |
| 17 | 4.1231 | β17 | 2.5713 |
| 18 | 4.2426 | 3β2 | 2.6207 |
| 19 | 4.3589 | β19 | 2.6684 |
| 20 | 4.4721 | 2β5 | 2.7144 |
| 21 | 4.5826 | β21 | 2.7589 |
| 22 | 4.6904 | β22 | 2.8020 |
| 23 | 4.7958 | β23 | 2.8439 |
| 24 | 4.8990 | 2β6 | 2.8845 |
| 25 | 5 | 5 | 2.9240 |
| 26 | 5.0990 | β26 | 2.9625 |
| 27 | 5.1962 | 3β3 | 3 |
| 28 | 5.2915 | 2β7 | 3.0366 |
| 29 | 5.3852 | β29 | 3.0723 |
| 30 | 5.4772 | β30 | 3.1072 |
A worked example you can check on paper
Find β72. The perfect squares that divide 72 are 4, 9 and 36; the largest is 36, so 72 = 36 x 2 and β72 = β36 x β2 = 6β2.
For the decimal, 72 lies between 64 and 81, so the root is between 8 and 9. Start at 8: the average of 8 and 72/8 = 9 is 8.5. The average of 8.5 and 72/8.5 = 8.4706 is 8.4853. One more pass gives 8.485281, and 6 x 1.41421356 = 8.4852814 confirms it.
Mistakes that catch people out
Splitting a sum. β(9 + 16) is β25 = 5, not β9 + β16 = 7. Roots split over multiplication, not addition.
Taking out a factor that is not the largest. β72 = 2β18 is true but not simplest; 18 still contains 9, giving 6β2.
Forgetting the negative root. xΒ² = 49 has two answers, 7 and -7, even though β49 on its own means 7.
Rounding too early. Rounding β2 to 1.41 before multiplying by 100 gives 141 instead of 141.42; keep the exact form until the end.
Popular calculations
Each opens the calculator with that root worked out, and the number can be changed once it is open.
Square roots that are not whole numbers
Perfect squares
Cube roots
Fourth and fifth roots
Negative numbers
Decimals
How this was checked
The engine was tested in Node against known values, such as β2 = 1.414213562373095 to 15 places and β123456789 = 11111.111061, and every root was raised back to its power to return the original number. Simplified forms were checked by hand, including 6β2 for 72, 3β2 for 54 and β2/2 for 0.5, along with perfect squares, cubes, zero and negative numbers.
Every address in the lists above was then opened and checked for a distinct title, the right root, charts that render, and clean CSV, image and print output containing only that number, at desktop, tablet and phone widths.
Common questions
What is the square root of 2?
About 1.414213562373095. It is irrational, so the decimal never ends or repeats, and β2 is its exact form. It is the length of the diagonal of a square with sides of 1.
How do you simplify a square root?
Find the largest perfect square that divides the number, write the number as that square times what is left, and take the root of the square. β72 = β(36 x 2) = 6β2, and β50 = β(25 x 2) = 5β2.
How do you find a square root by hand?
Start with a guess, then replace it with the average of the guess and the number divided by the guess. For β10, start at 3: the average of 3 and 10/3 is 3.1667, and the average of 3.1667 and 10/3.1667 is 3.1623, already correct to four decimal places.
What is the square root of a negative number?
There is no real square root, because squaring any real number gives a positive result. It is an imaginary number written with i, where iΒ² = -1: β-16 = 4i and β-1 = i. Cube roots of negative numbers are real, so β-8 = -2.
What is the difference between a square root and a cube root?
A square root multiplied by itself gives the number; a cube root multiplied by itself three times gives it. β64 = 8 because 8 x 8 = 64, while β64 = 4 because 4 x 4 x 4 = 64.
Is a square root always smaller than the number?
Only for numbers above 1. For numbers between 0 and 1 the root is larger: β0.25 = 0.5 and β0.01 = 0.1. The roots of 0 and 1 are the numbers themselves.
Related Calculators and Reference
- Quadratic Formula Calculator: equations whose roots need a square root.
- Prime Factorisation Calculator: the factors behind a simplified root.
- Distance and Pythagoras Calculator: the square root at work in lengths.
- Scientific Calculator: roots inside longer expressions.
- Log Calculator: logs and antilogs, the other side of powers and roots.
- Complex Number Calculator: square roots of negative numbers, and every root of a complex number.
- Wolfram MathWorld: Square Root: definitions and properties.