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Square Root Calculator

Find the square root of any number, with the simplified radical form, a perfect square check and the steps.

Results

ItemValue

Steps

StepWorking
Decimal answers are rounded to 10 significant digits. Negative numbers have no real square root, so the calculator shows the imaginary answer using i, where i² = −1.

Quick Answer

The square root of a number is the value that, multiplied by itself, gives that number. √81 = 9, because 9 × 9 = 81. For numbers that are not perfect squares, simplify by taking out square factors: √72 = √(36 × 2) = 6√2 ≈ 8.4853.

Last updated: 2 November 2026 ·How we calculate this: square root and radical simplification

Table of Contents

How This Square Root Calculator Works

Enter any number. The calculator gives the square root as a decimal and, for whole numbers, as a simplified radical such as 6√2. It also says whether the number is a perfect square and shows how the radical was simplified.

Quick explanation: √x × √x = x. Every positive number has two square roots, one positive and one negative.

  • Perfect squares: 1, 4, 9, 16, 25 and so on have whole-number roots.
  • Simplified radical: the exact answer, with square factors taken out.
  • Negative numbers: have imaginary roots, such as √−9 = 3i.

How to Use the Square Root Calculator

  1. Enter any number, whole or decimal.
  2. Press Calculate.
  3. Read the decimal square root.
  4. For whole numbers, read the simplified radical and whether it is a perfect square.
  5. Follow the steps to see how the radical was simplified.

Core Rules

  • Definition: √x = y means y² = x
  • Product rule: √(a × b) = √a × √b
  • Quotient rule: √(a ÷ b) = √a ÷ √b
  • As a power: √x = x^(1/2)

Key Terms

  • Square root: a number that gives the original number when multiplied by itself.
  • Perfect square: a whole number whose square root is a whole number.
  • Radical: the √ symbol, or an expression that uses it.
  • Radicand: the number under the root sign.
  • Imaginary unit (i): the number whose square is −1.

Worked Example

Simplify √180.

  • Find the largest square factor: 180 = 36 × 5
  • √180 = √36 × √5 = 6√5
  • As a decimal, 13.4164

Real-Life Examples

  • A square room: a square room of 196 sq ft has sides of √196 = 14 ft.
  • A TV diagonal: with a 3 ft by 4 ft screen, the diagonal is √(9 + 16) = 5 ft.
  • Standard deviation: it is the square root of the variance, so a variance of 25 gives 5.
  • Algebra homework: √48 simplifies to 4√3, and √200 to 10√2.

Square Root vs Cube Root

  • Square root: the number that, times itself, gives x. √64 = 8.
  • Cube root: the number that, used three times, gives x. ∛64 = 4.
  • Negatives: a negative number has no real square root, but it does have a real cube root: ∛−8 = −2.
  • More roots: use our Exponent Calculator with a fraction power, such as 1/3.

Square Roots of Common Numbers

NumberSquare rootSimplified
21.4142√2
31.7321√3
82.82842√2
123.46412√3
184.24263√2
204.47212√5
325.65694√2
507.07115√2
728.48536√2
989.89957√2
1001010
1441212

Tips for Square Roots

  • Learn the first squares: knowing 1² to 15² makes estimates quick.
  • Estimate: √50 is between 7 and 8, since 49 < 50 < 64.
  • Use prime factors: pairs of the same prime come out of the root.
  • Exact vs decimal: keep the radical form in algebra, and use decimals for measurements.

Common Mistakes to Avoid

  • Adding under the root. √(9 + 16) is 5, but √9 + √16 is 7.
  • Forgetting the negative root. x² = 25 has two answers: 5 and −5.
  • Not fully simplifying. √72 = 6√2, not 3√8.
  • Expecting a real root of a negative. √−9 is 3i, an imaginary number.
  • Rounding too early. Keep the radical form until the final step.

Frequently Asked Questions

About 1.4142135624. It cannot be written as an exact fraction.

12, because 12 × 12 = 144.

Find the largest perfect square that divides the number, take its root outside, and leave the rest inside.

Not in real numbers. With imaginary numbers, √−16 = 4i.

Only for numbers above 1. For numbers between 0 and 1, the root is larger: √0.25 = 0.5.

Estimate between two perfect squares, then refine. For √50, try 7.1: 7.1² = 50.41, so √50 is a little under 7.1.

0, because 0 × 0 = 0.

About 7.0711, or 5√2 in simplest radical form.

Because a negative times a negative is positive. Both 5 and −5 square to 25. The √ symbol means the positive one.

No. √2 is irrational, so its decimal goes on forever without repeating.

Multiply the numbers under the roots: √3 × √12 = √36 = 6.

Only like roots can be added: 2√3 + 5√3 = 7√3. Simplify first, as √12 + √3 = 2√3 + √3 = 3√3.

Yes. √x = x^(1/2).

References & Sources

SourceUsed ForLink
Wikipedia, Square root Definition and properties. https://en.wikipedia.org/wiki/Square_root
Wikipedia, Nth root Roots and radicals. https://en.wikipedia.org/wiki/Nth_root