Enter Terms
| Sign | Coefficient | Index n | Radicand b | Action |
|---|
Tip: Even indexes require non‑negative radicands for real results; odd indexes allow negative radicands.
Results
Exact simplified sum:
Numeric total (approx):
Unique like-term groups:
| Group | Index n | Radicand r | Combined coefficient |
|---|
Steps
Formula used
Each term is of the form a·√[n](b). Factor the radicand as b = p^n · r, where p is the product of prime powers that are perfect n‑th powers, and r is the remainder. Then:
a·√[n](b) = (a·p)·√[n](r)
Like terms share the same index n and remainder r after simplification; their coefficients are then added.
How to use this calculator
- Enter terms as sign, coefficient, index
n, and radicandb. - Click Compute to simplify and combine like radicals instantly.
- Review exact symbolic form and numeric approximation.
- Download a CSV or a nicely formatted PDF report.
Example data table
| # | Sign | Coeff | Index n | Radicand b |
|---|---|---|---|---|
| 1 | + | 3 | 2 | 8 |
| 2 | + | 2 | 2 | 18 |
| 3 | − | 1 | 2 | 2 |
| 4 | + | 1.5 | 3 | 54 |
Use “Load example” to populate the input table with these rows.
FAQs
After simplifying
b into p^n·r, like terms have the same index n and the same remainder r. Their coefficients are then added together.Odd indexes (e.g., cube roots) allow negative radicands. Even indexes require non‑negative radicands for real outputs.
They’re rounded to a default precision of six decimals on screen. The CSV preserves more raw precision.
Set the index
n to 3 for cube roots, 4 for fourth roots, and so on.This tool focuses on simplifying and adding radicals in the numerator. Denominator rationalization is out of scope for this calculator.
Yes. The Steps section logs each factorization and simplification, then shows how like terms were combined.
Common radical simplifications
| Expression | Factorization | Simplified exact form |
|---|---|---|
| √8 | 8 = 4·2 = 2²·2 | 2√2 |
| √18 | 18 = 9·2 = 3²·2 | 3√2 |
| √50 | 50 = 25·2 = 5²·2 | 5√2 |
| ∛54 | 54 = 27·2 = 3³·2 | 3∛2 |
| ∛16 | 16 = 8·2 = 2³·2 | 2∛2 |
| ∜32 | 32 = 16·2 = 2⁴·2 | 2∜2 |
Like-term grouping examples
| Input terms | After simplification | Groups | Exact sum | Approx. |
|---|---|---|---|---|
| 3√8 + 2√18 − √2 | 3·2√2 + 2·3√2 − 1·√2 | √2 group only | (6+6−1)√2 = 11√2 | ≈ 15.556 |
| 5∛16 − 2∛54 | 5·2∛2 − 2·3∛2 | ∛2 group only | (10−6)∛2 = 4∛2 | ≈ 5.0397 |
| 4∜32 + √50 | 4·2∜2 + 5√2 | Two groups (∜2, √2) | 8∜2 + 5√2 | ≈ 17.414 |
Index and sign rules (real outputs)
| Index n | Parity | Allowed radicand signs | Valid example | Invalid example |
|---|---|---|---|---|
| 2 | Even | ≥ 0 only | √18 = 3√2 | √(-2) not real |
| 3 | Odd | Any | ∛(-8) = -2 | — |
| 4 | Even | ≥ 0 only | ∜32 = 2∜2 | ∜(-2) not real |