Square Root Calculator
Calculate principal square roots with exact rational checks, simplified radical form, square intervals, and decimal approximations.
Last Updated: May 2026
Use a nonnegative integer, finite decimal, or fraction.
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. |
The square root symbol returns the nonnegative real root.
Perfect-square numerators and denominators return exact rational roots.
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.
Checked by Jitendra Kumar
Square Root Calculator is checked for formula labels, source links, and result limits.
Jitendra Kumar, Founder & Editorial Standards Lead. Updated May 2026. Scope: math calculators.
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
Related Calculators
Perfect Square Calculator
Check whether a whole number is a perfect square, then compare integer square roots and nearby squares.
Use Perfect Square CalculatorSimplifying Radicals Calculator
Simplify square roots, cube roots, and nth radicals by pulling perfect-power factors outside.
Use Simplifying Radicals CalculatorRadical Calculator
Simplify radicals, calculate nth roots, check exact rational roots, and compare radical forms with decimal approximations.
Use Radical CalculatorRoot Calculator
Calculate square roots, cube roots, and nth roots with exact rational checks and decimal approximations.
Use Root CalculatorRelated Guides
Fractions, Roots & Radicals Guide
Use this for subtracting fractions, Egyptian fractions, square roots, cube roots, nth roots, radicals, simplifying radicals, rationalizing denominators, perfect squares, and perfect cubes.
Read guide
Math Formulae Page: 440 K-12 Formulas
Use this as a broad formula reference across arithmetic, algebra, geometry, trigonometry, statistics, matrices, vectors, and calculus readiness.
Read guideSources & References
- 1.Khan Academy - Square roots(Accessed May 2026)
- 2.OpenStax - Square roots and cube roots(Accessed May 2026)
- 3.Wikipedia - Square root(Accessed May 2026)