Square Root Calculator
The square root of a number x is the value that, when multiplied by itself, equals x. This calculator finds both the principal (positive) and negative square roots of any non-negative real number. It also displays the result in simplified radical form, which is useful in algebra and higher mathematics. For example, the square root of 72 simplifies to 6 times the square root of 2 (6√2), because 72 = 36 × 2 and 36 is a perfect square. The calculator expresses this as sqrt(72) = 6 × sqrt(2). Square roots are fundamental in solving quadratic equations, calculating distances, and understanding geometric properties. This tool helps you understand the structure of square roots and their simplified forms.
Square root formula
√x = number that, when multiplied by itself, equals x
If a^2 = x, then a = √x or a = -√x
√x = x^(1/2)
Simplified form: factor out perfect squares
Perfect squares reference
| 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 | 20 | 400 |
| 7 | 49 | 25 | 625 |
| 8 | 64 | 30 | 900 |
| 9 | 81 | 50 | 2,500 |
| 10 | 100 | 100 | 10,000 |
Square root calculator: frequently asked questions
What is a square root?
The square root of a number x is a number that, when multiplied by itself, equals x. For example, the square root of 16 is 4, because 4 × 4 = 16. The square root symbol is sqrt or the radical sign: sqrt(x) or √x.
Can a square root be negative?
Mathematically, every positive number has two square roots: one positive and one negative. For example, both 4 and -4 are square roots of 16 because 4^2 = 16 and (-4)^2 = 16. This calculator shows both the positive and negative square roots.
What does simplified radical form mean?
Simplified radical form expresses a square root by factoring out perfect squares. For example, sqrt(72) = sqrt(36 × 2) = 6 × sqrt(2). This simplification is useful in mathematics and makes calculations easier.
What is the square root of a negative number?
The square root of a negative number is not a real number. It is an imaginary number involving i (the imaginary unit, where i^2 = -1). For example, sqrt(-4) = 2i. This calculator works with real numbers only.
What is the difference between sqrt and exponent notation?
The square root of x is the same as x^(1/2). Similarly, the cube root of x is x^(1/3). These are equivalent ways of expressing roots: sqrt(x) = x^(1/2).
Official sources
- Square roots: NIST Special Publication 330.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.