Radical Form Calculator
Simplify an nth root into radical form. Enter index n and radicand x, then see a·√[n]{b} and numeric evaluation immediately.
Equation Preview
Helping Notes
Fields mirror common radical/surd simplifiers: index n and radicand x. The result factors out perfect nth powers only. :contentReference[oaicite:1]{index=1}
If x = 0, the result is 0. Even index with negative x is invalid; odd index supports negative radicands.
Results
Simplified Radical
Outside Coefficient (a)
Remaining Radicand (b)
What Is a Radical Form Calculator?
A Radical Form Calculator rewrites numbers and algebraic expressions into simplified surd (radical) form. It extracts perfect powers from under the radical, rationalizes denominators, and converts between radicals and rational exponents. This is handy in algebra, geometry, trigonometry, and exam prep when exact, root‑based answers are preferred over decimals. Enter any square root, cube root, or nth root expression—such as , , or —and the tool returns a fully simplified, exact result along with intermediate steps. You can also choose whether to rationalize denominators and whether to keep absolute‑value bars when required by principal root conventions.
About the Radical Form Calculator
The engine factors integers to pull out perfect powers (e.g., ), applies product/quotient rules for radicals, and uses conjugates to remove radicals from denominators. It supports general indices , irreducible surds, and mixed symbolic inputs. When variables appear, it respects the principal square‑root rule and the identity . It can also convert rational exponents to radicals and back, using . If an expression contains multiple radicals, it combines like radicals and simplifies nested forms when algebraically valid. Throughout, it keeps exact arithmetic and only rounds when you request a decimal approximation.
Product rule:
Quotient rule:
Extracting perfect powers:
Rational exponents:
Rationalizing:
Combining radicals:
How to Use This Radical Form Calculator
- Enter your expression (e.g., , , or ).
- Select options: simplify, combine like radicals, and rationalize denominators. Choose the index if not a square root.
- Compute to see step‑by‑step factorization, extraction of perfect powers, and any conjugate multiplication for rationalization.
- Copy the final exact form; optionally toggle a decimal approximation for a quick numerical check.
Examples
- Simplify: .
- Combine & reduce: .
- Rationalize: .
- Cube root: .
- Variable (principal root): .
- Rational exponent: .
Formula Snippets Ready for Rendering
FAQs
What does “simplest radical form” mean?
No perfect powers remain under the radical, denominators are rational (if chosen), and like radicals are combined.
When do I need absolute values?
Because for real numbers; principal square roots are nonnegative.
Can the tool handle cube roots and higher?
Yes. Use the general index . For odd , absolute values are unnecessary.
How do I rationalize denominators with binomials?
Multiply numerator and denominator by the conjugate, e.g., .
Will decimals convert to radicals automatically?
Terminating decimals convert to rationals first; repeating decimals may be recognized and expressed as fractions, then simplified.
Can it combine unlike radicals?
Only like radicals with the same radicand and index combine directly (e.g., ).
What about nested radicals?
Some can be simplified by identities; otherwise they remain exact. The calculator attempts safe algebraic reductions.
Does it support variables?
Yes—keeps expressions symbolic and preserves principal root conventions and domain notes where relevant.
Why doesn’t my answer match my calculator?
Decimal calculators approximate; radical form is exact. Compare by squaring or evaluating numerically as a check.
Can I switch between radical and exponent forms?
Yes. Use and the reverse .