Square Root Calculator
Instantly find the square root of any positive number, complete with simplified radical forms, decimal precision, and perfect square references.
Calculate Any Square Root
Enter a positive number below to instantly calculate its square root, view its simplified radical form, and discover its relationship to nearby perfect squares.
Math Estimator
Get precise decimal results and simplified radical breakdowns.
How it Works
Understanding square root calculations
Input Validation
The tool first checks that your input is a valid, non-negative real number, as negative numbers require complex mathematics.
Decimal Calculation
Using high-precision native math functions, we calculate the exact decimal approximation to your chosen number of decimal places.
Radical Simplification
The algorithm finds the largest perfect square factor of your number to express the root in its simplest radical form (e.g., √12 = 2√3).
Contextual Data
We identify if the number is a perfect square and show the nearest perfect squares to help you estimate roots mentally in the future.
Perfect Squares Table
A quick reference guide for the first 20 perfect squares and their exact square roots.
| Integer (n) | Square (n²) | Square Root (√n) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 11 | 121 | 11 |
| 12 | 144 | 12 |
| 13 | 169 | 13 |
| 14 | 196 | 14 |
| 15 | 225 | 15 |
| 16 | 256 | 16 |
| 17 | 289 | 17 |
| 18 | 324 | 18 |
| 19 | 361 | 19 |
| 20 | 400 | 20 |
Math FAQ
Answers to the most frequently asked questions about square roots, radicals, and perfect squares.
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 × 4 = 16. It is denoted by the radical symbol √.
To simplify a square root, find the largest perfect square factor of the number. For example, to simplify √12, the largest perfect square factor is 4. You can rewrite it as √(4 × 3) = √4 × √3 = 2√3.
In the set of real numbers, no. Multiplying any real number by itself always results in a positive number or zero. However, in complex mathematics, the square root of a negative number involves the imaginary unit ‘i’, where i = √-1.
A perfect square is an integer that is the square of another integer. For example, 25 is a perfect square because it is 5 × 5. The square root of a perfect square is always a whole number with no decimal remainder.