Simplify Radical Calculator
Reduce a square root to its simplest exact form — √72 becomes 6√2.
A radical is simplified when no perfect square remains under the root.
How the simplify radical calculator works
A radical is simplified when no perfect square remains under the root. Factor out the largest square: 72 = 36 × 2, so √72 = 6√2.
The exact form is worth keeping. 6√2 is precise where 8.485 is rounded, and it stays precise through further algebra — which is why exams ask for surd form and why it survives in geometry, where roots of two and three appear constantly.
A coefficient in front multiplies the extracted part: 3√72 becomes 18√2.
Formula: √(a²b) = a√b
Worked examples
| Inputs | Simplified | Note |
|---|---|---|
| √72 | 6√2 | 6√2 |
| √144 — a perfect square | 12 | 12 |
| 3√72 | 18√2 | 18√2 |
FAQFrequently asked questions
How do I simplify √72?
Find the largest square factor — 36 — and take its root outside: √72 = √(36×2) = 6√2.
Why keep surd form?
Because it is exact. 6√2 stays precise through further algebra; 8.485 has already lost information.
What if the number is already prime?
Nothing comes out — √13 is already simplified.
How do I know a number is a perfect square?
Its simplified form has nothing left under the root, as with √144 = 12.
Can I simplify the root of a negative?
Not as a real number. √−9 is 3i, which needs complex numbers.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.