# Square Root Calculator > The square root of x is the number that gives x when multiplied by itself, and pulling out perfect-square factors simplifies it: √72 = √(36 × 2) = 6√2, about 8.49. The calculator also finds cube and nth roots, shows the prime factors and handles negative numbers, decimals and fractions. - Interactive calculator: https://calculatorswithsteps.com/square-root-calculator - Category: Math & Statistics Calculators - Last updated: 2026-11-10 - Publisher: Calculators With Steps (calculatorswithsteps.com) ## How to use the square root calculator 1. Choose **square root**, **cube root** or **nth root**. For nth root, enter the index n (4 for a fourth root, 5 for a fifth root and so on). 2. Type the number. Whole numbers, decimals (0.25), fractions (1/2) and negative numbers (−27) all work. The answer comes as a decimal to 10 places and, where possible, in simplified radical form. You also see whether the number is a perfect square or cube, which two perfect squares it sits between, and its prime factorisation. ## Square root formula and notation The square root of x is the non-negative number that gives x when multiplied by itself: **√x = y where y × y = x and y ≥ 0** - The **cube root** ∛x is the number y with y³ = x. - The **nth root** ⁿ√x is the number y with yⁿ = x. - Roots are fractional powers: **√x = x^(1/2)**, ∛x = x^(1/3) and ⁿ√x = x^(1/n). The [exponent calculator](https://calculatorswithsteps.com/exponent-calculator) handles powers like 8^(2/3). Every positive number has two square roots, one positive and one negative (both 12 and −12 square to 144). The √ symbol means the positive one, called the principal square root. ## How to simplify a square root Simplest radical form means no perfect square is left under the root sign. The calculator does it with prime factorisation: | Step | √72 | √180 | | --- | --- | --- | | Prime factors | 72 = 2³ × 3² | 180 = 2² × 3² × 5 | | Pull out pairs | (2² × 3²) × 2 = 36 × 2 | (2² × 3²) × 5 = 36 × 5 | | Take roots | √36 × √2 | √36 × √5 | | Simplified | 6√2 | 6√5 | | Decimal | 8.4852813742 | 13.416407865 | For cube roots, pull out groups of three instead of pairs: 54 = 2 × 3³, so ∛54 = 3∛2. This is also how the [quadratic formula calculator](https://calculatorswithsteps.com/quadratic-formula-calculator) writes exact answers like 1 ± √2. ## Finding a square root by hand The Babylonian (or Heron's) method improves a guess with each round: 1. Pick a guess g. For √10, g = 3 because 3² = 9 is close. 2. Divide: 10 ÷ 3 ≈ 3.3333. 3. Average: (3 + 3.3333) ÷ 2 ≈ 3.1667. 4. Repeat with the new guess: (3.1667 + 10 ÷ 3.1667) ÷ 2 ≈ 3.16228. The true value is 3.1622776602, so two rounds already give five correct decimal places. Each round roughly doubles the number of correct digits. ## Negative numbers and imaginary roots A negative number has no real square root. In algebra and engineering it is written with the imaginary unit i, defined by i² = −1, so √(−72) = √72 × i = 6√2 i. Odd roots of negative numbers are ordinary real numbers: ∛(−27) = −3, because (−3)³ = −27. Even roots higher than two, such as the fourth root of −16, have no real value, and the calculator says so. ## Perfect squares and perfect cubes from 1 to 20 | n | n² (perfect square) | n³ (perfect cube) | | --- | --- | --- | | 1 | 1 | 1 | | 2 | 4 | 8 | | 3 | 9 | 27 | | 4 | 16 | 64 | | 5 | 25 | 125 | | 6 | 36 | 216 | | 7 | 49 | 343 | | 8 | 64 | 512 | | 9 | 81 | 729 | | 10 | 100 | 1,000 | | 11 | 121 | 1,331 | | 12 | 144 | 1,728 | | 13 | 169 | 2,197 | | 14 | 196 | 2,744 | | 15 | 225 | 3,375 | | 16 | 256 | 4,096 | | 17 | 289 | 4,913 | | 18 | 324 | 5,832 | | 19 | 361 | 6,859 | | 20 | 400 | 8,000 | Knowing these makes estimating easy. √72 must be between 8 and 9 because 72 sits between 64 and 81, and ∛100 is between 4 and 5 because 100 sits between 64 and 125. Square roots also turn up whenever you measure a diagonal, which is why the [Pythagorean theorem calculator](https://calculatorswithsteps.com/pythagorean-theorem-calculator) uses them for every answer. ## Frequently asked questions ### What is the square root of 2? The square root of 2 is about 1.4142135624. It is irrational, so its decimal never ends or repeats, and √2 is already in simplest radical form because 2 is prime. ### What is the square root of 144? The square root of 144 is 12, because 12 × 12 = 144. Its prime factorisation is 2⁴ × 3², and halving each exponent gives 2² × 3 = 12. ### How do you simplify a square root? Factor the number, pull out every pair of equal factors, and leave the rest under the root. 72 = 2³ × 3² = 36 × 2, so √72 = √36 × √2 = 6√2, which is about 8.4852813742. ### What is the square root of a negative number? There is no real square root of a negative number, because any real number squared is zero or positive. Using the imaginary unit i, where i² = −1, √(−72) = 6√2 i, which is about 8.4852813742i. ### How do you find a cube root? Look for groups of three equal prime factors. 54 = 2 × 3³, so ∛54 = 3∛2, about 3.7797631497. Cube roots of negative numbers are real and negative: ∛(−27) = −3. ### What is the square root of 0 and of 1? √0 = 0 and √1 = 1, because 0 × 0 = 0 and 1 × 1 = 1. They are the only numbers that equal their own square root. ### How do you calculate a square root without a calculator? Use the Babylonian method: guess, divide the number by your guess, and average the two. For √10, start at 3, then (3 + 10 ÷ 3) ÷ 2 ≈ 3.1667, and one more round gives about 3.16228, already correct to five decimal places. ### How do you find the square root of a fraction or decimal? Take the root of the top and bottom separately. √0.25 = √(1/4) = 1/2 = 0.5. When the bottom is not a perfect square, rationalise it: √(1/2) = √2/2, about 0.7071067812. ## Sources - [Wikipedia – Square root](https://en.wikipedia.org/wiki/Square_root) - [Wolfram MathWorld – Square Root](https://mathworld.wolfram.com/SquareRoot.html) - [Wikipedia – nth root](https://en.wikipedia.org/wiki/Nth_root)