{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"simplify-radical"},"result":{"entity_type":"calculator","id":"simplify-radical","calculator_id":"simplify-radical","canonical_url":"https://tttkmbb.com/math/simplify-radical","name":"Radical Simplifier","title":"Radical Simplifier – Simplify Square Roots and Cube Roots to a√b Form","category":"math","category_name":"Math","tool_name":"calculate_simplified_radical","featured_mcp_tool":false,"description":"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.","use_when":"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.","inputs":[{"name":"radicand","label":"Radicand n","type":"integer","required":true,"min":-1000000000000,"max":1000000000000,"description":"Integer under the root sign (negative only with index 3).","example":50},{"name":"index","label":"Root index","type":"integer","required":false,"default":2,"min":2,"max":3,"description":"2 for a square root, 3 for a cube root.","example":2},{"name":"coefficient","label":"Coefficient","type":"integer","required":false,"default":1,"min":-1000000,"max":1000000,"description":"Optional multiplier in front of the radical, e.g. 3 for 3√50.","example":1}],"outputs":[{"name":"simplified","label":"Simplified radical","type":"string","decimals":4,"description":"Simplest form such as '5√2', '3∛2', '-2∛3' or a plain integer when the radicand is a perfect power."},{"name":"outside_coefficient","label":"Coefficient outside","type":"integer","decimals":0,"description":"The integer in front of the simplified radical (including the input coefficient and sign)."},{"name":"remaining_radicand","label":"Radicand inside","type":"integer","decimals":0,"description":"The integer left under the root sign (1 when the root is exact)."},{"name":"decimal","label":"Decimal value","type":"number","decimals":8,"description":"coefficient × n^(1/index)."},{"name":"is_perfect_power","label":"Perfect square/cube","type":"boolean","decimals":4,"description":"true when the root is an integer."},{"name":"factorization","label":"Prime factorization","type":"string","decimals":4,"description":"Prime-power form of |n|, e.g. '50 = 2 × 5^2'."},{"name":"expression","label":"Working","type":"string","decimals":4,"description":"e.g. '√50 = √(25 × 2) = 5√2'."}],"input_schema":{"type":"object","properties":{"radicand":{"description":"Integer under the root sign (negative only with index 3).","type":"integer","minimum":-1000000000000,"maximum":1000000000000,"examples":[50]},"index":{"description":"2 for a square root, 3 for a cube root.","type":"integer","minimum":2,"maximum":3,"default":2,"examples":[2]},"coefficient":{"description":"Optional multiplier in front of the radical, e.g. 3 for 3√50.","type":"integer","minimum":-1000000,"maximum":1000000,"default":1,"examples":[1]}},"additionalProperties":false,"required":["radicand"]},"output_schema":{"type":"object","properties":{"simplified":{"description":"Simplest form such as '5√2', '3∛2', '-2∛3' or a plain integer when the radicand is a perfect power.","type":"string"},"outside_coefficient":{"description":"The integer in front of the simplified radical (including the input coefficient and sign).","type":"integer"},"remaining_radicand":{"description":"The integer left under the root sign (1 when the root is exact).","type":"integer"},"decimal":{"description":"coefficient × n^(1/index).","type":"number"},"is_perfect_power":{"description":"true when the root is an integer.","type":"boolean"},"factorization":{"description":"Prime-power form of |n|, e.g. '50 = 2 × 5^2'.","type":"string"},"expression":{"description":"e.g. '√50 = √(25 × 2) = 5√2'.","type":"string"}}},"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","sources":[{"name":"Wikipedia – Nth root – simplified form of a radical expression","url":"https://en.wikipedia.org/wiki/Nth_root#Simplified_form_of_a_radical_expression","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Square root","url":"https://en.wikipedia.org/wiki/Square_root","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wolfram MathWorld – Radical","url":"https://mathworld.wolfram.com/Radical.html","type":"reference","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"√50","inputs":{"radicand":50,"index":2,"coefficient":1},"expected":{"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"},"url":"https://tttkmbb.com/api/v1/calculate/simplify-radical?radicand=50&index=2&coefficient=1"},{"name":"∛54","inputs":{"radicand":54,"index":3},"expected":{"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"},"url":"https://tttkmbb.com/api/v1/calculate/simplify-radical?radicand=54&index=3"}],"faq":[{"q":"What does 'simplest radical form' mean?","a":"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."},{"q":"Why is the square root of a negative number rejected?","a":"√−n is imaginary (i√n), not a real radical; only odd roots of negative numbers are real, e.g. ∛−54 = −3∛2."}],"tags":["simplify radical","simplify square root","simplest radical form","cube root simplify","perfect square factor","surd"],"related":[{"calculator_id":"nth-root","reason":"Numeric value of any root, including higher indices."},{"calculator_id":"prime-factorization","reason":"The factorization used to extract perfect powers."},{"calculator_id":"quadratic-equation","reason":"Roots that often need to be simplified."}],"links":{"html":"https://tttkmbb.com/math/simplify-radical","markdown":"https://tttkmbb.com/math/simplify-radical.md","json":"https://tttkmbb.com/math/simplify-radical.json","api":"https://tttkmbb.com/api/v1/calculate/simplify-radical","schema":"https://tttkmbb.com/api/v1/calculators/simplify-radical","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T01:51:38Z"}