Simplify a Radical
Enter a coefficient, a whole-number radicand, and the root degree. The calculator extracts perfect powers, shows the exact simplified form, gives a decimal approximation, and prepares exportable output.
Example Data Table
| Expression | Factor Insight | Simplified Form | Decimal Approx. |
|---|---|---|---|
| √72 | 72 = 36 × 2 | 6√2 | 8.485281 |
| 3√50 | 50 = 25 × 2 | 15√2 | 21.213203 |
| ∛54 | 54 = 27 × 2 | 3∛2 | 3.779763 |
| 2∜80 | 80 = 16 × 5 | 4∜5 | 5.981395 |
| ∛(-16) | -16 = -(8 × 2) | -2∛2 | -2.519842 |
Formula Used
Core idea:
If a radicand contains a perfect nth power, you can move that factor outside the radical.
General rule:
a × √[n]{b} = a × √[n]{kn × r} = a × k × √[n]{r}
Where: b = kn × r, k is the extracted factor, and r is the remaining radicand.
Prime-factor method:
Write the absolute radicand as prime powers. For each prime exponent e, move out ⌊e / n⌋ copies and leave e mod n copies inside.
For odd roots, a negative radicand keeps a real answer because the negative sign can move outside the radical.
How to Use This Calculator
- Enter the coefficient in front of the radical. Use 1 if there is no visible coefficient.
- Type the radicand as a whole number. Negative values are allowed only for odd roots.
- Set the root degree. Use 2 for square roots, 3 for cube roots, and higher values for nth roots.
- Choose the decimal precision for the approximation field.
- Click Simplify Radical to place the result above the form.
- Review the exact form, decimal value, factorization, and step-by-step explanation.
- Use the CSV and PDF buttons to export the result summary.
Frequently Asked Questions
1) What does simplifying a radical mean?
It means removing every perfect square, cube, or higher perfect power from inside the radical. The result keeps an exact form with the smallest possible remaining radicand.
2) Why must the radicand be a whole number?
This calculator is designed for exact radical simplification using factorization. Whole-number radicands let the tool detect perfect powers reliably and provide clean symbolic answers.
3) Can this calculator simplify cube roots and fourth roots?
Yes. Change the root degree from 2 to 3, 4, or another value up to 12. The calculator then groups prime factors according to that degree.
4) What happens with negative radicands?
Negative radicands work only for odd roots in real numbers. For example, ∛(-16) is valid, but √(-16) is not a real result, so the calculator blocks it.
5) Why do some answers lose the radical completely?
That happens when the full radicand is a perfect nth power. Then every factor comes outside, leaving only a regular number with no remaining radical part.
6) What is the difference between exact and decimal output?
The exact output keeps the radical form, such as 6√2. The decimal output converts the same value into an approximation, which is useful for measurement and comparison.
7) How is the extracted perfect power found?
The calculator factors the absolute radicand into primes, groups equal factors by the root degree, and multiplies those complete groups outside the radical.
8) What do the export buttons include?
The CSV and PDF exports include the main result summary, such as the original expression, simplified exact form, decimal approximation, extracted power, and remaining radicand.