{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"nth-root"},"result":{"entity_type":"calculator","id":"nth-root","calculator_id":"nth-root","canonical_url":"https://tttkmbb.com/math/nth-root","name":"Nth Root Calculator","title":"Nth Root Calculator – Square, Cube and Higher Roots (Real Roots of Negative Numbers)","category":"math","category_name":"Math","tool_name":"calculate_nth_root","featured_mcp_tool":false,"description":"Computes the principal n-th root of a number (value^(1/n)), returning the real negative root for odd n and the second, negative root for even n.","use_when":"You need √x, ∛x or any integer-degree root, including odd roots of negative numbers.","do_not_use_when":"You need a fractional power such as x^(2/3) (use exponent) or the solutions of a quadratic (use quadratic-equation).","inputs":[{"name":"value","label":"Radicand","type":"number","required":true,"min":-1e+300,"max":1e+300,"description":"The number under the root sign. Negative values are allowed only for odd n.","example":16},{"name":"n","label":"Root degree n","type":"integer","required":true,"min":1,"max":1000,"description":"2 for square root, 3 for cube root, etc.","example":4}],"outputs":[{"name":"root","label":"Principal root","type":"number","decimals":8,"description":"The real n-th root with the same sign as value."},{"name":"negative_root","label":"Negative root","type":"number","decimals":8,"description":"−root: the second real root, present only for even n and positive value."},{"name":"verification","label":"root^n","type":"number","decimals":6,"description":"The root raised back to the n-th power (should equal value)."},{"name":"expression","label":"Expression","type":"string","decimals":4,"description":"The root as text, e.g. '16^(1/4) = 2'."}],"input_schema":{"type":"object","properties":{"value":{"description":"The number under the root sign. Negative values are allowed only for odd n.","type":"number","minimum":-1e+300,"maximum":1e+300,"examples":[16]},"n":{"description":"2 for square root, 3 for cube root, etc.","type":"integer","minimum":1,"maximum":1000,"examples":[4]}},"additionalProperties":false,"required":["value","n"]},"output_schema":{"type":"object","properties":{"root":{"description":"The real n-th root with the same sign as value.","type":"number"},"negative_root":{"description":"−root: the second real root, present only for even n and positive value.","type":"number"},"verification":{"description":"The root raised back to the n-th power (should equal value).","type":"number"},"expression":{"description":"The root as text, e.g. '16^(1/4) = 2'.","type":"string"}}},"formula":"root = |value|^(1/n) × sign(value); for even n and value > 0 the roots are ±root","method":"Roots that are exact integers are snapped to the integer when |value|^(1/n) is within 10^-9 of it and the integer power reproduces value exactly.","sources":[{"name":"Wikipedia – Nth root","url":"https://en.wikipedia.org/wiki/Nth_root","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wolfram MathWorld – nth Root","url":"https://mathworld.wolfram.com/nthRoot.html","type":"reference","retrieved_at":"2026-09-23"}],"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":"Fourth root of 16","inputs":{"value":16,"n":4},"expected":{"root":2,"negative_root":-2,"verification":16,"expression":"16^(1/4) = 2"},"url":"https://tttkmbb.com/api/v1/calculate/nth-root?value=16&n=4"},{"name":"Fifth root of −32","inputs":{"value":-32,"n":5},"expected":{"root":-2,"verification":-32},"url":"https://tttkmbb.com/api/v1/calculate/nth-root?value=-32&n=5"}],"faq":[{"q":"Why does the square root of a negative number fail?","a":"Even roots of negative numbers are imaginary (√−4 = 2i); only odd roots of negatives are real, e.g. ∛−27 = −3."},{"q":"Why are there two answers for an even root?","a":"Both 2 and −2 satisfy x⁴ = 16. The principal root is the positive one; negative_root gives the other."}],"tags":["nth root","square root","cube root","radical","root of a number"],"related":[{"calculator_id":"exponent","reason":"Inverse operation and fractional exponents."},{"calculator_id":"logarithm","reason":"Roots via logarithms: log(root) = log(value) / n."},{"calculator_id":"quadratic-equation","reason":"Roots of second-degree equations."}],"links":{"html":"https://tttkmbb.com/math/nth-root","markdown":"https://tttkmbb.com/math/nth-root.md","json":"https://tttkmbb.com/math/nth-root.json","api":"https://tttkmbb.com/api/v1/calculate/nth-root","schema":"https://tttkmbb.com/api/v1/calculators/nth-root","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:23:14Z"}