Education
Square Root Calculator
Enter a number to get the square root as a decimal at the precision you pick, plus a simplified radical form and a perfect square check. Optional imaginary mode handles negative inputs.
The number to take the square root of. · e.g. 72
Square root
8.4853
√72 ≈ 8.4853
√72 simplifies to 6√2. As a decimal, √72 ≈ 8.4853.
Examples
√25
= 5 · perfect square
√72
= 6√2 ≈ 8.4853
√2
≈ 1.4142 · irrational
√(-9)
= 3i · imaginary
How it works
The square root of a number x is the value y that satisfies y × y = x. The principal square root is the non-negative solution.
Square root formula
√x = y where y × y = x
Simplified radical (integer input)
√x = a · √b
where a² is the largest perfect-square factor of x and b = x / a².
Negative real inputs do not have a real square root. With imaginary mode enabled, the calculator returns √x · i for negative x, where i is the imaginary unit and i² = -1.
What is a square root calculator?
A square root calculator finds the value that, when multiplied by itself, gives the input number. For a perfect square, the answer is an integer. For other non-negative numbers, the answer is an irrational decimal that the calculator rounds to the precision you select. For non-negative integer inputs, it can also return a simplified radical form like 6√2 for √72.
How the square root calculator works
Enter a number, and the calculator:
- Computes the principal (positive) square root.
- Rounds the decimal value to your chosen precision.
- For a non-negative integer input, factors out the largest perfect square and reports the simplified radical form.
- Flags whether the input is a perfect square (its root is an integer).
- Handles negative inputs as either undefined in the reals or as an imaginary number, depending on the toggle.
Square root formula
The defining relationship is √x = y when y × y = x and y ≥ 0. Both y and -y square to x, so a positive number has two square roots, but the radical symbol refers to the non-negative one.
Simplifying a square root
To simplify √x for a non-negative integer x, find the largest perfect-square factor of x and pull its root outside the radical:
- √72 = √(36 × 2) = √36 · √2 = 6√2
- √50 = √(25 × 2) = 5√2
- √98 = √(49 × 2) = 7√2
- √12 = √(4 × 3) = 2√3
The simplified form is exact and is the standard answer in algebra and geometry classes. The decimal form is useful for measurement and physical answers.
Perfect squares
A perfect square is a non-negative integer whose square root is also an integer. The first dozen perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and 121. The calculator flags whether the input is a perfect square so you can quickly tell whether the root is rational.
Square root of a negative number
In the real numbers, the square root of a negative number is undefined. There is no real value y with y² = -9, because any real number squared is non-negative.
In the complex numbers, the imaginary unit i is defined by i² = -1, which gives √(-x) = √x · i for any positive x. For example, √(-9) = 3i and √(-2) = √2 · i ≈ 1.4142i. Enable the imaginary toggle to see this form.
Square root examples
- √4 = 2, because 2 × 2 = 4
- √25 = 5, because 5 × 5 = 25
- √72 = 6√2 ≈ 8.4853
- √2 ≈ 1.4142 (irrational)
- √0.25 = 0.5, because 0.5 × 0.5 = 0.25
- √(-9) = 3i in the complex numbers
Common mistakes
- Forgetting that a positive number has two square roots. The symbol √ returns only the non-negative one; for the full solution to y² = x, write y = ±√x.
- Treating √(a + b) as √a + √b. The square root does not distribute over addition. For example, √(9 + 16) = √25 = 5, not 3 + 4 = 7.
- Trying to take a real square root of a negative number. The result is not a real number; use the imaginary form or report no real solution.
- Leaving an unsimplified radical in a math answer. In algebra and geometry, the expected form is the simplified radical (for example, 6√2, not √72).
Where the square root shows up
The square root is everywhere in math and physics. It appears in the Pythagorean theorem, where the hypotenuse of a right triangle is c = √(a² + b²); in the distance formula between two points; in the quadratic formula, where the discriminant lives under the radical; in standard deviation, defined as the square root of the variance; and in many physics formulas, from kinematics to oscillation periods.
For specific uses, see the Pythagorean theorem calculator and the quadratic formula calculator.
Related tools
- Quadratic formula calculator solves ax² + bx + c = 0 using the discriminant.
- Pythagorean theorem calculator finds the missing side of a right triangle.
- Percentage calculator runs the three standard percent calculations.
- All education calculators.
Note. Decimal results are rounded for display; the calculator uses standard floating-point precision internally. Simplified radical form is computed for non-negative integer inputs only.
Frequently asked questions
The square root of a number x is the value y such that y times y equals x. In symbols, √x = y means y × y = x. Every positive number has two square roots, one positive and one negative, but the symbol √ refers to the principal (positive) square root.
For a perfect square, find the integer that multiplies by itself to give the input. For non-perfect squares, the result is irrational, so we report it as a decimal rounded to the chosen number of places. The calculator uses the standard floating-point square root and rounds the displayed value for readability.
Simplifying a square root means writing it as outside × √inside, where inside has no remaining perfect-square factors. For example, √72 simplifies to 6√2, because 72 = 36 × 2 and √36 = 6. The simplified form is exact, while the decimal form is rounded.
A perfect square is a non-negative integer whose square root is also an integer. Examples include 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. If the input is a perfect square, the calculator returns the exact integer result and flags it as a perfect square.
Not as a real number. The square root of a negative number is not defined within the real numbers. If you enable imaginary numbers, the calculator returns the i form: √(-x) = √x · i for any positive x. For example, √(-9) = 3i.
The square root of 0 is 0, because 0 × 0 = 0. Zero is the only number whose principal square root equals itself (other than 1, where √1 = 1).
The square root of 1 is 1, because 1 × 1 = 1. Like 0, it is a fixed point of the square root function in the non-negative reals.
No. The square root of 2 is irrational, meaning it cannot be written as a ratio of two integers. Its decimal expansion (about 1.4142135...) never terminates and never repeats. The calculator shows it to the precision you select.
Squaring a number multiplies it by itself: 5² = 5 × 5 = 25. The square root reverses that operation: √25 = 5. They are inverse operations on the non-negative reals.
The square root appears in many geometric formulas, including the Pythagorean theorem (c = √(a² + b²)), the distance formula, the diagonal of a square (d = s√2), and the radius of a circle from its area (r = √(A/π)).
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