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Square Root Calculator

Calculate principal square roots, identify perfect squares, and simplify radicals into exact a√b expressions.

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How to Use the Square Root Calculator – Radicals & Simplification

1Enter a positive number into the radicand input box.
2Read the principal square root and simplified radical expression.
3Check whether your number is a perfect square.
4Review the prime factor breakdown and step-by-step working.

Formula & How It Works

Radical Simplification & Factor Extraction

√x = √(a² × b) = a√b

For any positive real number x, the principal square root √x is the non-negative number y such that y² = x. To simplify radicals into a√b form, extract the largest perfect square factor a² from x, yielding a√b where b has no square factors.

Worked Examples

Perfect Square Integer

Input: 144

Explanation: 12 × 12 = 144, making 144 an exact perfect square integer.

Result: 12 (Perfect Square: 12² = 144)

Radical Simplification Example

Input: 72

Explanation: 72 factors into 36 × 2. Extracting √36 yields 6√2.

Result: 6√2 ≈ 8.485281

Prime Radicand

Input: 17

Explanation: 17 has no square factors greater than 1, so the radical cannot be simplified.

Result: √17 ≈ 4.123106

Decimal Radicand

Input: 0.25

Explanation: Square roots of fractions between 0 and 1 are larger than the radicand.

Result: 0.5 (Since 0.5² = 0.25)

Frequently Asked Questions

What is a principal square root?▼

While every positive number has two square roots (one positive and one negative), the principal square root refers specifically to the non-negative root denoted by √x.

Can you calculate the square root of a negative number?▼

In the real number system, negative numbers do not have real square roots. In complex numbers, √(-x) = i√x, where i is the imaginary unit.

What is a perfect square?▼

A perfect square is an integer that is the square of an integer (such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 144).

How do you simplify square roots by hand?▼

Find the largest perfect square factor of the number, take its square root outside the radical sign, and leave the remainder inside.

What is the square root of 2?▼

√2 is an irrational number approximately equal to 1.41421356... that cannot be written as a simple fraction.

Is this calculator free and browser-based?▼

Yes, all square root arithmetic runs 100% in your local browser without transmitting data.

Last reviewed & updated: 2026-11-01

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