Math Tool
Square Root Calculator
Calculate square roots, cube roots, and any nth root instantly. Get simplified radical form, step-by-step solutions, prime factorization, and a visual square root curve โ all in one tool.
5 ร 5 = 25
โn = n1/2
โn = n1/3
Root Calculator
Calculate any root instantly
Root type
Enter number
Decimal places
Supports positive numbers, decimals, and fractions. Negative numbers return imaginary results (i notation).
โ144 = 12 (exact)
Step-by-step solution
โ144Square Root Curve
โx from 0 to 100 โ your value is highlighted
Perfect squares reference table
| n | nยฒ | โnยฒ | nยณ | โnยณ |
|---|
How it works
What is a square root?
The square root of a number is a value that, when multiplied by itself, gives the original number. Written as โn, it is one of the most fundamental operations in mathematics โ appearing in algebra, geometry, physics, statistics, and everyday problem-solving.
Every positive number has two square roots: a positive and a negative. For example, both 5 and โ5 are square roots of 25 because 5ยฒ = 25 and (โ5)ยฒ = 25. By convention, โ25 refers to the positive root (called the principal root).
Square roots are closely connected to the Pythagorean theorem. If you know two sides of a right triangle, the third side always involves a square root. For example, a triangle with legs of 3 and 4 has a hypotenuse of โ(3ยฒ + 4ยฒ) = โ25 = 5. Try the Pythagorean Theorem Calculator for full triangle calculations.
Methods
How to calculate square roots
Prime Factorization
Break the number into prime factors, pair them up, and bring one of each pair outside the radical. Best for perfect squares and simplifying.
โ144 = โ(2โดร3ยฒ) = 2ยฒร3 = 12 Long Division Method
A manual algorithm that works digit by digit, similar to long division. Gives accurate decimal results for any number.
โ2 โ 1.41421356... Estimation Method
Find the two perfect squares the number is between, then interpolate. For example, โ50 is between โ49=7 and โ64=8, closer to 7.
โ50 โ 7.07 Related math tools
More calculators that use square roots
Square roots appear in many geometry calculations. The most common is the Pythagorean Theorem Calculator, which uses square roots to find the missing side of a right triangle โ essential for construction, navigation, and physics problems. Another geometry tool where square roots appear is the Circumference Calculator, specifically when you need to find the radius of a circle from its area using r = โ(A รท ฯ). Both calculators show step-by-step work so you can follow the full solution.
Examples
Square root examples
Example 1 โ Perfect square
โ144 = 12 because 12 ร 12 = 144. This is an exact result with no decimals โ 144 is a perfect square.
Example 2 โ Simplify radical
โ48 = โ(16 ร 3) = 4โ3 โ 6.928. The largest perfect square factor of 48 is 16, so โ16 = 4 comes outside.
Example 3 โ Cube root
โ216 = 6 because 6 ร 6 ร 6 = 216. Cube roots are used in volume calculations โ finding the side length of a cube from its volume.
Example 4 โ Irrational root
โ2 โ 1.41421356โฆ This is an irrational number โ its decimal expansion never ends or repeats. It is the length of the diagonal of a unit square.
Guide
How to use this square root calculator
- Choose root type. Click Square (โ), Cube (โ), 4th root, or Custom for any nth root.
- Enter your number. Type any positive number, decimal, or large integer.
- For custom roots, enter the root degree (e.g. 5 for 5th root).
- Select decimal places. Choose 2, 4, 6, or 10 decimal places for precision.
- View results. See the decimal result, simplified radical, squared value, and whether it is a perfect square.
- Read the steps. The step-by-step solution shows prime factorization or the calculation method used.
- Copy or share. Use the copy buttons or share directly on WhatsApp and Facebook.
FAQ
Square Root Calculator FAQ
What is a square root?
The square root of a number n is a value x such that x ร x = n. For example, the square root of 25 is 5 because 5 ร 5 = 25. It is written as โn.
How do I simplify a square root?
To simplify โn, find the largest perfect square factor of n. For example, โ48 = โ(16 ร 3) = โ16 ร โ3 = 4โ3. This is called simplifying the radical.
What is a perfect square?
A perfect square is a number whose square root is a whole integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Their square roots are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
What is the square root of a negative number?
The square root of a negative number is an imaginary number. โ(-n) = iโn, where i is the imaginary unit. For example, โ(-9) = 3i.
What is a cube root?
The cube root of n is the value x such that x ร x ร x = n. It is written as โn. For example, โ27 = 3 because 3 ร 3 ร 3 = 27.
How is square root used in real life?
Square roots appear in the Pythagorean theorem (finding the hypotenuse), calculating distances, areas, standard deviations in statistics, physics formulas, and engineering calculations.