{
  "id": "root",
  "version": "38511e5c59bd",
  "status": "published",
  "name": "Root Calculator",
  "question": "What is the nth root of a number?",
  "summary": "Finds the nth root of a number, or the number or the degree from the other two values.",
  "category": "math",
  "subcategory": "arithmetic",
  "url": "https://www.acalculator.org/math/root-calculator",
  "markdown": "https://www.acalculator.org/math/root-calculator.md",
  "kind": "formulas",
  "method": "r = ⁿ√x = x^(1/n): the principal real nth root of x.",
  "assumptions": [
    "Real numbers only: an even root of a negative number has no answer.",
    "An odd whole degree gives the negative root of a negative number: ∛−8 = −2.",
    "The principal root: √x is the positive root, so √100 is 10, not −10."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "n": {
        "title": "Degree (n)",
        "description": "Which root to take: 2 for a square root, 3 for a cube root.",
        "type": "number"
      },
      "value": {
        "title": "Number (x)",
        "description": "The number to take the root of.",
        "type": "number"
      },
      "answer": {
        "title": "Root",
        "description": "The principal nth root of the number.",
        "type": "number"
      }
    }
  },
  "solve": {
    "fillAny": 2,
    "variables": [
      "n",
      "value",
      "answer"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "degree": {
      "label": "Degree (n)",
      "description": "Which root to take: 2 for a square root, 3 for a cube root.",
      "format": "number"
    },
    "value": {
      "label": "Number (x)",
      "description": "The number to take the root of.",
      "format": "number"
    },
    "root": {
      "label": "Root",
      "description": "The principal nth root of the number.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "n": 2,
      "value": 100
    },
    "outputs": {
      "root": 10
    },
    "text": "The root of degree 2 of 100 is 10."
  },
  "examples": [
    {
      "given": {
        "n": 2,
        "value": 100
      },
      "expect": {
        "answer": 10
      },
      "source": "hand calculation in content.mdx: 10² = 100; OpenStax, Intermediate Algebra 2e, §8.1 Simplify Expressions with Roots. https://openstax.org/books/intermediate-algebra-2e/pages/8-1-simplify-expressions-with-roots"
    },
    {
      "given": {
        "n": 4,
        "value": 81
      },
      "expect": {
        "answer": 3
      },
      "source": "hand calculation in content.mdx: 3⁴ = 81; OpenStax, Intermediate Algebra 2e, §8.1 Simplify Expressions with Roots. https://openstax.org/books/intermediate-algebra-2e/pages/8-1-simplify-expressions-with-roots"
    },
    {
      "given": {
        "n": 3,
        "value": -8
      },
      "expect": {
        "answer": -2
      },
      "source": "hand calculation in content.mdx: (−2)³ = −8, so ∛−8 = −2; OpenStax, Intermediate Algebra 2e, §8.1 Simplify Expressions with Roots. https://openstax.org/books/intermediate-algebra-2e/pages/8-1-simplify-expressions-with-roots"
    },
    {
      "given": {
        "n": 5,
        "value": 10000000000
      },
      "expect": {
        "answer": 100
      },
      "source": "hand calculation in content.mdx: 100⁵ = 10¹⁰; OpenStax, Intermediate Algebra 2e, §8.1 Simplify Expressions with Roots. https://openstax.org/books/intermediate-algebra-2e/pages/8-1-simplify-expressions-with-roots"
    },
    {
      "given": {
        "value": 1024,
        "answer": 2
      },
      "expect": {
        "n": 10
      },
      "source": "hand calculation in content.mdx: 2¹⁰ = 1,024, so n = log₂ 1,024 = 10; OpenStax, Intermediate Algebra 2e, §8.1 Simplify Expressions with Roots. https://openstax.org/books/intermediate-algebra-2e/pages/8-1-simplify-expressions-with-roots"
    },
    {
      "given": {
        "n": 3,
        "answer": 2
      },
      "expect": {
        "value": 8
      },
      "source": "hand calculation in content.mdx: x = 2³ = 8; OpenStax, Intermediate Algebra 2e, §8.1 Simplify Expressions with Roots. https://openstax.org/books/intermediate-algebra-2e/pages/8-1-simplify-expressions-with-roots"
    },
    {
      "given": {
        "n": 2,
        "value": 2
      },
      "expect": {
        "answer": 1.4142135623730951
      },
      "source": "hand calculation in content.mdx: √2 = 1.41421356…; Python 3: 2 ** 0.5 = 1.4142135623730951; OpenStax, Intermediate Algebra 2e, §8.1 Simplify Expressions with Roots. https://openstax.org/books/intermediate-algebra-2e/pages/8-1-simplify-expressions-with-roots"
    }
  ],
  "sources": [
    "OpenStax, Intermediate Algebra 2e, §8.1 Simplify Expressions with Roots. https://openstax.org/books/intermediate-algebra-2e/pages/8-1-simplify-expressions-with-roots"
  ],
  "related": [],
  "changelog": [
    {
      "date": "2026-09-28",
      "note": "A degree of 0 names the rule."
    },
    {
      "date": "2026-09-29",
      "note": "Worked examples name a published reference for their rule."
    }
  ]
}