Calculator
White themeEnter terms as coefficient and radicand. All terms share the same index n.
| Coefficient | Radicand | Action |
|---|---|---|
Form: (a·√[n]{x}) / (b·√[n]{y}). Leave x or y as 1 to omit radical.
Server time: 2026-01-25 19:01
Example Data
Click a row to load the values into the calculator.
| Mode | n | Coefficient a | Radicand x | More |
|---|---|---|---|---|
| Simplify | 2 | 1 | 72 | √72 → 6√2 |
| Simplify | 3 | 2 | 54 | 2∛54 → 6∛2 |
| Combine | 2 | — | — | 3√72 − 2√18 + 5√50 |
| Rationalize | 3 | 1 | 1 | (√[3]{1})/(2√[3]{5}) |
| Simplify | 4 | 1 | 1296 | √[4]{1296} → 6√[4]{6} |
Formulas Used
- Product: √[n]{a}·√[n]{b} = √[n]{ab}.
- Quotient: √[n]{a}/√[n]{b} = √[n]{a/b}, b ≠ 0.
- Simplify: write a = ∏ pᵉ; extract ∏ p^{⌊e/n⌋} outside, leave ∏ p^{e mod n} inside.
- Rationalize: for b·√[n]{y} in denominator, multiply top and bottom by √[n]{y^{n−1}} so denominator becomes b·y.
- Add/Subtract: combine terms sharing identical inside radicand after simplification.
How to Use
- Pick a tab matching your task: simplify, combine, or rationalize.
- Enter index n ≥ 2 and the required coefficients/radicands.
- Press the action button to see the simplified form and steps.
- Export your session steps with CSV or PDF buttons below.
Common Perfect Powers
For k ≤ 10: squares 1,4,9,16,25,36,49,64,81,100; cubes 1,8,27,64,125,216,343,512,729,1000.
Worked Examples Table
| # | Mode | Input | Output | Key Steps |
|---|
Radical Identities and Operations
| Operation | Expression | Result | Note |
|---|---|---|---|
| Product rule | √[n]{a} · √[n]{b} | √[n]{ab} | Valid for a,b ≥ 0, integer n ≥ 2. |
| Quotient rule | √[n]{a} / √[n]{b} | √[n]{a/b} | b ≠ 0. Keep radicand non‑negative for real results. |
| Exponent form | √[n]{a} | a^{1/n} | Useful for algebraic manipulation and proofs. |
| Extraction | √[n]{p^{kn+r}} | p^{k} √[n]{p^{r}} | k = ⌊e/n⌋, r = e mod n for prime powers. |
| Like radicals | c₁√[n]{m} + c₂√[n]{m} | (c₁+c₂)√[n]{m} | Combine only after full simplification inside. |
Perfect Powers Reference
| k | k² (square) | k³ (cube) | k⁴ (fourth power) |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 16 |
| 3 | 9 | 27 | 81 |
| 4 | 16 | 64 | 256 |
| 5 | 25 | 125 | 625 |
| 6 | 36 | 216 | 1296 |
| 7 | 49 | 343 | 2401 |
| 8 | 64 | 512 | 4096 |
| 9 | 81 | 729 | 6561 |
| 10 | 100 | 1000 | 10000 |
Rationalization Multipliers Guide
| Index n | Denominator form | Multiply numerator and denominator by | Denominator becomes |
|---|---|---|---|
| 2 | b·√{y} | √{y} | b·y |
| 3 | b·√[3]{y} | √[3]{y²} | b·y |
| 4 | b·√[4]{y} | √[4]{y³} | b·y |
| 5 | b·√[5]{y} | √[5]{y⁴} | b·y |