Square Root Calculator

Name: Square Root Calculator
Author: Jitendra Kumar

Calculate principal square roots with exact rational checks, simplified radical form, square intervals, and decimal approximations.

Last Updated: May 2026

Radicand

Use a nonnegative integer, finite decimal, or fraction.

Decimal precision

Whole number from 0 to 12.

Decimal approximation

8.485281

Exact root

Not rational

Simplified radical

6 sqrt(2)

Square interval

8^2 < 72 < 9^2

Item	Value	Meaning
Original square root	sqrt(72)	Parsed as a integer.
Exact radicand	72	Finite decimals and fractions are reduced before root checks.
Exact root	Irrational or not rational	Shown when numerator and denominator are perfect squares.
Simplified radical	6 sqrt(2)	Perfect-square factors are pulled outside the square root.
Square interval	8^2 < 72 < 9^2	Neighboring integer squares for whole-number radicands.
Decimal approximation	8.485281	Rounded to 6 decimal places.

Perfect-Square Steps

Step	Factorization	Meaning
Numerator square factor	72 = 36 x 2	6 comes outside because 6^2 = 36.
Denominator square factor	1 = 1 x 1	1 comes outside the denominator.
Principal root	Nonnegative real root	The square root symbol returns the principal square root.

Principal root

The square root symbol returns the nonnegative real root.

Exact checks

Perfect-square numerators and denominators return exact rational roots.

Simplified form

Irrational square roots stay exact as simplified radicals plus a decimal estimate.

Real Square Root Notice

This calculator returns real principal square roots. It does not list complex square roots for negative radicands.

Reviewed For Methodology, Labels, And Sources

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Reviewed By

Jitendra Kumar, Founder & Editorial Standards Lead, reviews methodology, labels, assumptions, and trust-sensitive publishing decisions for this topic area.

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Topic Ownership

Sales tax and tax-sensitive estimate tools, Education and GPA planning calculators, Health, protein, and screening-formula pages, Platform-wide publishing standards and methodology

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Methodology & Updates

Page updated May 2026. Trust-critical pages are reviewed when official rates or rules change. Evergreen calculator guides are checked on a recurring quarterly or annual cycle depending on topic volatility.

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How to Use the Square Root Calculator

Enter a nonnegative integer, finite decimal, or fraction. Then choose the number of decimal places for the displayed approximation.

Review the exact root when available, the simplified radical form when the root is irrational, and the square interval for whole-number radicands.

Step 1: Enter the radicand
Use a nonnegative value such as 144, 72, 2, 81/16, or 0.0625.
Step 2: Set decimal precision
Choose 0 to 12 decimal places for the approximation.
Step 3: Check exactness
If the radicand is a perfect-square rational value, the exact root is shown.
Step 4: Review the radical form
Use simplified radical form for exact algebra and the decimal for estimation.

How This Square Root Calculator Works

The calculator converts finite decimals and fractions into exact rational values. It checks the numerator and denominator for perfect-square factors.

Perfect-square factors are moved outside the radical. If no inside factor remains, the square root is shown as an exact integer or reduced fraction.

For whole-number radicands, the interval row shows the neighboring integer squares that bound the answer.

Square Root Guide

Square Root Rules

Concept	Example	Meaning
Definition	sqrt(x) = y means y^2 = x	The principal square root reverses squaring.
Perfect square	sqrt(144) = 12	Because 12^2 = 144.
Simplified radical	sqrt(72) = 6 sqrt(2)	Pull perfect-square factors outside.
Fraction root	sqrt(81/16) = 9/4	Check numerator and denominator exactly.
Irrational root	sqrt(2) ~= 1.414214	Some square roots need decimal approximations.
Real domain	sqrt(-9)	No real result for negative radicands.

Square Root Examples

Radicand	Result	Notes
144	12	Perfect square.
72	6 sqrt(2)	Simplified radical form.
2	sqrt(2)	Irrational exact form.
81/16	9/4	Exact rational square root.
0.0625	0.25	Finite decimal perfect square.

Exact Form vs Decimal Form

Exact radical form is preferred for algebra because it avoids rounding. Decimal approximations are useful for measurement, estimation, graphing, and quick comparison.

Keep the research moving with Perfect Square Calculator, Simplifying Radicals Calculator, Radical Calculator, and Root Calculator.

Frequently Asked Questions

A square root of a number is a value that gives the original number when multiplied by itself. The square root symbol returns the principal nonnegative root.

Yes. It pulls perfect-square factors outside the radical, such as sqrt(72) = 6 sqrt(2).

Yes. Finite decimals and fractions are converted to exact rational values before checking for exact roots and simplified radical form.

This page works in the real-number domain. Negative radicands have complex square roots, which are not listed here.

The Square Root Calculator finds exact, simplified, and decimal square roots. The Perfect Square Calculator only checks whether a whole number is an integer square.