Root Calculator

Name: Root Calculator
Author: Jitendra Kumar

Calculate square roots, cube roots, and nth roots, then compare exact rational roots with rounded decimal approximations.

Last Updated: May 2026

Radicand

Integer, finite decimal, or fraction.

Root index

Whole number from 2 to 20.

Decimal precision

Whole number from 0 to 12.

Decimal root

12.00

Exact root

Expression

sqrt(144)

Power interval

12^2 = 144

Item	Value	Meaning
Expression	sqrt(144)	Principal real square root.
Input type	integer	Finite decimals and fractions are converted exactly.
Exact radicand	144	Reduced value used for exact checks.
Perfect power?	Yes	Both numerator and denominator must be perfect powers.
Exact root	12	Shown when an exact rational root exists.
Decimal root	12.00	Rounded to 6 decimal places.
Power interval	12^2 = 144	Bounds the answer for whole-number radicands.
Check	12^2 = 144	Exact roots can be verified by raising them to the selected index.

Principal root

The calculator reports the principal real root for square roots and higher roots.

Exact checks

Perfect powers return exact rational roots. Other values keep a rounded decimal.

Approximation

Decimal precision controls rounding only; exact fraction checks use the full input.

Root identity

If sqrt(x) = y, then y raised to 2equals x. Exact roots include this check in the table above.

Real Root Notice

This calculator returns real roots. It does not list complex roots for negative radicands with even root indexes.

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 Root Calculator

Enter a radicand as an integer, finite decimal, or fraction. Then choose the root index: 2 for square root, 3 for cube root, or any whole number up to 20 for higher roots.

Use the exact root when it appears. If the root is irrational, use the decimal approximation and power interval to estimate the answer.

Step 1: Enter the radicand
Use a value such as 144, -125, 625, 81/16, or 0.008.
Step 2: Choose the root index
Use 2 for square roots, 3 for cube roots, or a larger whole number for nth roots.
Step 3: Set rounding precision
Choose 0 to 12 decimal places for the approximation.
Step 4: Check the result
Review the exact root, decimal root, and verification row.

How This Root Calculator Works

The calculator converts finite decimals and fractions into exact rational numbers. It then checks whether the numerator and denominator are perfect powers for the chosen index.

When the root is exact, the result is shown as an integer or reduced fraction. When the root is not rational, the calculator reports a rounded principal real root.

For nonnegative whole-number radicands, the power interval shows the two nearby integer powers that bracket the answer.

Root and Radicand Guide

Root Rules

Concept	Example	Meaning
Square root	sqrt(144) = 12	A number that squares back to the radicand.
Cube root	cuberoot(-125) = -5	Odd roots can have negative real answers.
Fourth root	root(4, 625) = 5	A number that raised to the fourth power gives the radicand.
Fraction root	sqrt(81/16) = 9/4	The numerator and denominator are checked exactly.
Irrational root	sqrt(2)	No exact rational root; use the decimal approximation.

Root Examples

Input	Result	Notes
144, index 2	12	Exact square root.
-125, index 3	-5	Exact negative cube root.
625, index 4	5	Exact fourth root.
81/16, index 2	9/4	Exact fraction root.
2, index 2	1.4142135624...	Irrational square root.

Exact Roots vs Decimal Roots

Exact roots are best for algebra because they avoid rounding. Decimal roots are useful for measurement, estimation, and quick comparisons when the exact value is irrational.

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

Frequently Asked Questions

A root calculator finds the value that must be raised to a selected power to return the radicand. For example, the square root of 144 is 12 because 12^2 = 144.

Yes. Enter any whole-number index from 2 to 20 to calculate square roots, cube roots, fourth roots, and higher real roots.

Yes. You can enter fractions such as 81/16. The calculator reduces the value exactly before checking whether the root is rational.

This calculator works in the real-number domain. Negative values have real odd roots, but they do not have real even roots.

The Root Calculator focuses on exact and decimal root values. The Radical Calculator also simplifies radical expressions by pulling perfect-power factors outside the radical.