Radical Calculator
Simplify radicals, calculate nth roots, check exact rational roots, and compare exact radical form with decimal approximations.
Last Updated: May 2026
Enter an integer, finite decimal, or fraction.
Whole number from 2 to 12.
Whole number from 0 to 12.
Decimal approximation
8.485281
Exact root
Not rational
Simplified radical
6 sqrt(2)
Radicand as fraction
72
| Item | Value | Meaning |
|---|---|---|
| Radicand | 72 | Parsed as a integer. |
| Root index | 2 | sqrt of the radicand. |
| Exact rational input | 72 | Finite decimals and fractions are simplified before root checks. |
| Perfect nth power? | No | Both numerator and denominator must be perfect nth powers. |
| Exact root | Irrational or not rational | Shown when the root is rational. |
| Simplified radical | 6 sqrt(2) | Nth-power factors are pulled outside the radical when practical. |
| Decimal approximation | 8.485281 | Rounded for display to 6 decimal places. |
| Check | Approximation only | Raising an exact root to the index returns the radicand. |
The simplified form keeps the answer exact when the root is irrational.
Perfect nth powers return rational roots; other values keep a radical and decimal.
Odd roots accept negative radicands. Even roots stay in the real-number domain.
If root(index, x) = y, then y raised to that index equals x. The check row shows that identity whenever the exact root is rational.
Real Radical Notice
This calculator focuses on real roots. It does not list complex roots for negative radicands with even root indexes.
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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.
How to Use the Radical Calculator
Enter a radicand as an integer, finite decimal, or fraction. Then set the root index: use 2 for square roots, 3 for cube roots, or another whole number up to 12.
Read the exact root when available, the simplified radical form when the root is irrational, and the rounded decimal approximation for measurement-style use.
Step 1: Enter the radicand
Use a value such as 72, 54, 162, 81/16, or -128.
Step 2: Choose the index
Use 2 for square roots, 3 for cube roots, or 4 through 12 for higher roots.
Step 3: Set decimal precision
Choose 0 to 12 decimal places for the displayed approximation.
Step 4: Compare exact and decimal forms
Use the simplified radical for exact math and the decimal for quick estimation.
How This Radical Calculator Works
The calculator parses finite decimals and fractions as exact rational values. It checks whether the numerator and denominator are perfect nth powers, which determines whether the root has an exact rational result.
If the root is not rational, the calculator searches for nth-power factors that can be pulled outside the radical. The remaining inside value stays under the radical sign.
Odd-index roots support negative radicands because they have real negative roots. Even-index roots require nonnegative radicands in the real-number system.
Radical and Nth Root Guide
Radical Rules
| Concept | Example | Meaning |
|---|---|---|
| Square root | sqrt(72) = 6 sqrt(2) | Pull perfect-square factors outside the radical. |
| Cube root | cuberoot(54) = 3 cuberoot(2) | Pull perfect-cube factors outside the radical. |
| Nth root | root(4, 162) = 3 root(4, 2) | Pull perfect fourth-power factors outside. |
| Fraction radicand | sqrt(81/16) = 9/4 | Both numerator and denominator are checked exactly. |
| Odd negative root | root(7, -128) = -2 | Odd-index real roots can be negative. |
| Even negative root | sqrt(-9) | No real result in this calculator. |
Radical Examples
| Input | Simplified Result | Notes |
|---|---|---|
| 72, index 2 | 6 sqrt(2) | Square-root simplification. |
| 54, index 3 | 3 cuberoot(2) | Cube-root simplification. |
| 162, index 4 | 3 root(4, 2) | Fourth-root simplification. |
| 81/16, index 2 | 9/4 | Exact rational square root. |
| -128, index 7 | -2 | Exact odd-index root. |
Exact Form vs Decimal Form
Exact radical form is usually preferred in algebra because it preserves the full value without rounding. Decimal approximations are useful when comparing sizes, checking reasonableness, or using a result in a measurement context.
Keep the research moving with Cube Root Calculator, Perfect Square Calculator, Scientific Calculator, and Prime Factorization Calculator.
Frequently Asked Questions
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Use Power Mod CalculatorSources & References
- 1.OpenStax - Square roots and cube roots(Accessed May 2026)
- 2.Khan Academy - Roots, exponents, and scientific notation(Accessed May 2026)
- 3.Wikipedia - Radical symbol(Accessed May 2026)