What is a square root?

The square root of a number x is a value r such that r² = x. Every positive number has two square roots: one positive and one negative. By convention, √ refers to the positive root.

Can you take the square root of a negative number?

Not in real numbers. The square root of a negative number is an imaginary number. For example, √(-4) = 2i, where i is the imaginary unit. This calculator handles non-negative inputs.

What is a perfect square?

A perfect square is a number whose square root is an integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. They are squares of 1, 2, 3, 4, ... respectively.

No. √2 is irrational, meaning it cannot be expressed as a fraction of two integers. Its decimal expansion (1.41421356...) never terminates or repeats.

How are square roots used in the Pythagorean theorem?

In a right triangle with legs a and b, the hypotenuse c = √(a² + b²). For a 3-4-5 triangle: c = √(9+16) = √25 = 5.

How do I simplify a square root?

Factor out perfect squares. √72 = √(36×2) = 6√2. Find the largest perfect square factor and take its root outside the radical.

Square Root Calculator

Calculate the square root of any number. Supports positive numbers, perfect squares detection, and decimal precision.

Number

Root Type

Square Root

12.0000000000

Perfect root

Verification

144.000000

12.000000^2

n Squared

20,736

144² = 20,736

n Cubed

2,985,984

144³ = 2,985,984

1 / √n

0.08333333

Reciprocal of root

Prime Factorization

2^4 × 3^2

Of 144

Nearby Perfect Squares

9² = 81

10² = 100

11² = 121

12² = 144

13² = 169

14² = 196

15² = 225

16² = 256

Planning notes, formulas, and examples

About the Square Root Calculator

The Square Root Calculator finds the square root of any non-negative number. The square root of x is the number that, when multiplied by itself, gives x. For example, √25 = 5 because 5 × 5 = 25.

Square roots appear throughout mathematics, science, and engineering: calculating distances (Pythagorean theorem), standard deviations in statistics, signal processing, and geometric calculations. It shows precise results and indicates whether the input is a perfect square.

The calculator handles perfect squares (giving exact integers), non-perfect squares (giving decimal results), and very large numbers. It also shows the squared value for verification.

When This Page Helps

While simple square roots like √9 = 3 are easy, roots of non-perfect squares require computation. It shows precise results for any value.

How to Use the Inputs

Enter a non-negative number.
The square root appears in the result panel.
Check if the number is a perfect square.
View the result to high decimal precision.
Verify by squaring the result.

Formula used

√x = x^(1/2)

The result r satisfies: r × r = x
Perfect square: if r is a whole number

Example Calculation

Result: 12

√144 = 12. Since 12 × 12 = 144, and 12 is a whole number, 144 is a perfect square.

Tips & Best Practices

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144..
√2 ≈ 1.4142 is irrational and never terminates.
The square root of a negative number is imaginary (√-1 = i).
For quick estimation: √50 is between √49=7 and √64=8, closer to 7.
Pythagorean theorem: c = √(a² + b²).

Square Roots in Geometry

The Pythagorean theorem, distance formula, and circle equations all involve square roots. Computing distances between points, finding diagonal lengths, and solving quadratic equations are everyday applications.

Estimating Square Roots

For mental math, bracket the number between two perfect squares. √50 is between 7 (√49) and 8 (√64). Since 50 is much closer to 49, √50 ≈ 7.07.

Square Roots and Standard Deviation

In statistics, standard deviation is the square root of variance. This makes it one of the most frequently used mathematical operations in data analysis.

Mastering this concept provides a strong foundation for advanced coursework in mathematics, statistics, and related quantitative disciplines.

Sources & Methodology

Last updated: February 8, 2026

Frequently Asked Questions

The square root of a number x is a value r such that r² = x. Every positive number has two square roots: one positive and one negative. By convention, √ refers to the positive root.