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Radical Simplifier

Writes √n or ∛n (optionally with a coefficient) in simplest radical form a√b by extracting perfect squares or cubes from the prime factorization of the radicand, and gives the decimal value.

When to use

You need √50 as 5√2, ∛54 as 3∛2, or want to know whether an integer is a perfect square or cube.

Do not use when: You only need the numeric value of a root (use nth-root), a root of a non-integer, or roots of degree 4 and higher.

Formula

n = Π pᵢ^eᵢ; ⁿ√n = (Π pᵢ^⌊eᵢ/k⌋) · ᵏ√(Π pᵢ^(eᵢ mod k)) for index k; for a negative radicand and k = 3 the sign moves outside

Inputs

Parameter	Type	Required	Description
`radicand`	integer	yes	Integer under the root sign (negative only with index 3). Range: ≥ -1000000000000, ≤ 1000000000000
`index`	integer	default 2	2 for a square root, 3 for a cube root. Range: ≥ 2, ≤ 3
`coefficient`	integer	default 1	Optional multiplier in front of the radical, e.g. 3 for 3√50. Range: ≥ -1000000, ≤ 1000000

Outputs

Output	Type	Description
`simplified`	string	Simplest form such as '5√2', '3∛2', '-2∛3' or a plain integer when the radicand is a perfect power.
`outside_coefficient`	integer	The integer in front of the simplified radical (including the input coefficient and sign).
`remaining_radicand`	integer	The integer left under the root sign (1 when the root is exact).
`decimal`	number	coefficient × n^(1/index).
`is_perfect_power`	boolean	true when the root is an integer.
`factorization`	string	Prime-power form of \|n\|, e.g. '50 = 2 × 5^2'.
`expression`	string	e.g. '√50 = √(25 × 2) = 5√2'.

Example

√50: {"radicand":50,"index":2,"coefficient":1} → {"simplified":"5√2","outside_coefficient":5,"remaining_radicand":2,"decimal":7.07106781,"is_perfect_power":false,"factorization":"50 = 2 × 5^2","expression":"√50 = √(25 × 2) = 5√2"}

∛54: {"radicand":54,"index":3} → {"simplified":"3∛2","outside_coefficient":3,"remaining_radicand":2,"decimal":3.77976315,"is_perfect_power":false,"factorization":"54 = 2 × 3^3","expression":"∛54 = ∛(27 × 2) = 3∛2"}

GET https://tttkmbb.com/api/v1/calculate/simplify-radical?radicand=50&index=2&coefficient=1

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/simplify-radical (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/simplify-radical · Markdown: https://tttkmbb.com/math/simplify-radical.md · JSON definition: https://tttkmbb.com/math/simplify-radical.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="simplify-radical"
OpenAPI operationId: calculate_simplified_radical
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Nth root – simplified form of a radical expression (reference)
Wikipedia – Square root (reference)
Wolfram MathWorld – Radical (reference)

FAQ

What does 'simplest radical form' mean?

The radicand contains no factor that is a perfect square (for √) or perfect cube (for ∛): √50 = √(25 × 2) = 5√2, because 25 is the largest square factor of 50.

Why is the square root of a negative number rejected?

√−n is imaginary (i√n), not a real radical; only odd roots of negative numbers are real, e.g. ∛−54 = −3∛2.

Related calculators

Nth Root Calculator — Numeric value of any root, including higher indices.
Prime Factorization Calculator — The factorization used to extract perfect powers.
Quadratic Equation Solver — Roots that often need to be simplified.