Inputs (white theme)
Operand A
Operand B (hidden for “Simplify”)
Tip: Even index requires non‑negative radicands. Odd index supports negatives.
Result
Exact form
Decimal value
Steps
Example Data Table
| # | Timestamp | Operation | Operand A | Operand B | Exact | Decimal |
|---|
Rows you add will appear here for export.
Formulas used
- Product rule: \u221A[n](a)\u22C5\u221A[n](b) = \u221A[n](ab), n\u22652.
- Quotient rule: \u221A[n](a)/\u221A[n](b) = \u221A[n](a/b), b\u22600.
- Unifying indices: \u221A[n](a)\u22C5\u221A[m](b) = \u221A[l](a^{l/n}b^{l/m}), where l = lcm(n,m).
- Simplifying: factor radicand: a = \u220F p_i^{e_i}. Pull outside: \u220F p_i^{\u230A e_i/n \u230B}; inside keeps p_i^{e_i \u2261 mod\ n}.
- Decimal evaluation: \u221A[n](a) = a^{1/n}. For negatives, only odd n permitted.
- Rationalizing: 1/(\u03BA\u221A[n](d)) \u00D7 \u221A[n](d)^{n-1}/\u221A[n](d)^{n-1} = \u221A[n](d)^{n-1}/(\u03BA d).
How to use this calculator
- Pick an operation: simplify, add, subtract, multiply, or divide.
- Enter each term as outside coefficient, index, and radicand.
- Set decimal precision. Toggle steps and auto‑simplify if desired.
- Press Calculate to see exact and decimal results.
- Click Add to table to save the result.
- Export the table via CSV or PDF when you are done.
FAQs
After simplification, terms must have the same index and the same inside radicand. Only their outside coefficients are combined algebraically.
Yes for odd indices (cube root, fifth root, etc.). For even indices, the radicand must be non‑negative to remain in real numbers.
It factors the radicand into primes, pulls perfect n-th powers outside the radical, and multiplies them into the outside coefficient.
The calculator uses the least common multiple of indices to rewrite both radicals with a common index, then applies product or quotient rules and simplifies.
If you tick “Rationalize denominator,” and the denominator is a single radical times a constant, it rewrites the expression to remove radicals from the denominator.
You can set precision from 0–12 decimal places. Exact forms remain symbolic; decimal values are rounded to your selected precision.
Common Radical Simplifications
| Expression | Simplified Exact Form | Decimal (≈) |
|---|---|---|
| √8 | 2√2 | 2.828427 |
| √12 | 2√3 | 3.464102 |
| √18 | 3√2 | 4.242640 |
| √50 | 5√2 | 7.071068 |
| ∛16 | 2∛2 | 2.519842 |
| ∛54 | 3∛2 | 3.779763 |
Use these as quick references when checking calculator outputs.
Domain & Validation Rules
- Even index: radicand must be ≥ 0 to stay in reals.
- Odd index: radicand may be negative; result keeps sign.
- Like radicals: same index and same simplified radicand.
- Division: denominator cannot be zero after simplification.
- Precision: decimals rounded to selected places (0–12).
| Check | Pass Example | Fail Example |
|---|---|---|
| Even index domain | √(9) is valid | √(-9) invalid in reals |
| Odd index domain | ∛(-8) = -2 | — |
| Like radicals (add/sub) | 2√8 + 3√18 → 7√2 | 2√2 + 3√3 stays as-is |
| Division by zero | √12 ÷ √3 = √4 | k·√r ÷ 0 is undefined |
© CodingAce — Basic Radical Operations Calculator