Square Root Calculator
Find square roots of any number.
Understanding Square Root Calculator
The Square Root Calculator finds the principal square root of a non-negative number. The square root of a number is the value that, when multiplied by itself, produces the original number. It is written using the radical symbol √. For example, √144 = 12 because 12 × 12 = 144.
Every positive real number has exactly one principal square root, which is defined to be non-negative. Although both 12 and −12 have a square of 144, the principal square root of 144 is 12. The negative value is the other solution of the equation x² = 144, but it is not represented by the principal radical √144.
Turning the inputs into a useful result
Square Root Calculator is intended to show the mathematical relationship behind the answer, not just return a number. Enter the quantities requested by the selected method—such as Number—and keep their units or mathematical meaning consistent. Where this calculator offers different shapes, methods, or forms, choose the one that matches the problem before entering the values.
A quick check before using the result
For a useful check, look at the units and the scale of the answer before accepting it. A formula can be evaluated correctly while an input is still inappropriate for the problem. Repeating the calculation with a simple test value or checking the result against the expected direction of change can help identify an input mistake.
About Square Root Calculator
The Square Root Calculator finds the principal square root of a non-negative number. The square root of a number is the value that, when multiplied by itself, produces the original number. It is written using the radical symbol √. For example, √144 = 12 because 12 × 12 = 144.
Every positive real number has exactly one principal square root, which is defined to be non-negative. Although both 12 and −12 have a square of 144, the principal square root of 144 is 12. The negative value is the other solution of the equation x² = 144, but it is not represented by the principal radical √144.
Some numbers are perfect squares. A perfect square is a number whose square root is an integer, such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and 144. Other positive numbers have irrational square roots. For example, √2 cannot be represented exactly as a terminating or repeating decimal and is approximately 1.41421356.
Square roots are closely related to exponents. The square root of a number can be written as a fractional exponent: √x = x1/2. More generally, fractional exponents provide a connection between powers and roots, which is useful throughout algebra, geometry, calculus, physics, and engineering.
Square roots are also important in geometry. For example, the Pythagorean theorem can be used to find the length of the hypotenuse of a right triangle, and the resulting expression often contains a square root. Square roots also appear in distance formulas, circle calculations, statistics, and many scientific equations.
In the real number system, the square root of a negative number is not a real number. For example, √−9 is not real because no real number multiplied by itself can produce −9. In the complex number system, however, √−9 can be represented as 3i, where i is the imaginary unit and i² = −1.
How the Square Root Calculator Works
Enter any non-negative number and select Calculate. The calculator determines its principal square root and displays the result with additional information that helps verify and understand the calculation.
√a = b
where b² = a
The calculator also checks whether the input is a perfect square. If the square root is an integer, the number is a perfect square. Otherwise, the calculator identifies it as a non-perfect square and provides its decimal approximation.
Square Root and Exponents
A square root can be expressed as a fractional exponent. This gives another equivalent way to calculate the same value.
For example, √144 can be written as 1441/2, which equals 12. This relationship is especially useful when simplifying algebraic expressions involving powers and roots.
Perfect Squares
A perfect square is an integer that can be written as the square of another integer. Recognizing perfect squares can make calculations easier and can sometimes allow a square root to be simplified exactly without using a decimal approximation.
2² = 4
3² = 9
4² = 16
5² = 25
10² = 100
12² = 144
Examples
√144 = 12
12 × 12 = 144
Result: 12
√50 ≈ 7.07106781
7.07106781² ≈ 50
The result is irrational and is therefore displayed as a decimal approximation.
√0 = 0
0² = 0
Square Root vs. Negative Square Root
It is important to distinguish between the principal square root and the solutions of a squared equation. The expression √25 refers to the principal square root and equals 5. However, the equation x² = 25 has two real solutions: x = 5 and x = −5.
x² = 25 → x = ±5
Square Root in Geometry
Square roots frequently occur when calculating distances and lengths. For a right triangle, the Pythagorean theorem states that a² + b² = c². Solving for the hypotenuse gives c = √(a² + b²).
a² + b² = c²
c = √(a² + b²)
Important Mathematical Considerations
This calculator works with real square roots. Therefore, negative inputs do not produce a real-number result. When a negative number is entered, the calculator reports that the square root is not real rather than displaying an invalid numerical value.
Decimal results are approximations when the exact square root is irrational. Increasing the number of displayed decimal places can provide a closer approximation, but an irrational number cannot be represented exactly by a finite decimal.
Common Uses
Square roots are used in algebra, geometry, trigonometry, statistics, physics, engineering, finance, computer science, and scientific calculations. Common applications include finding distances, calculating standard deviation, solving quadratic relationships, determining dimensions, and working with mathematical formulas involving powers and roots.
Disclaimer
This calculator is intended for informational purposes only. While we strive for accuracy, we cannot guarantee that every result is correct for every situation.