{
  "id": "simplifying-radicals",
  "version": "9f06f106ab39",
  "status": "published",
  "name": "Simplifying Radicals Calculator",
  "question": "Need help simplifying radicals?",
  "summary": "Writes a square root, cube root, or any nth root of a whole number in simplest radical form, such as √72 = 6√2, with the prime factors and the steps.",
  "category": "math",
  "subcategory": "algebra",
  "url": "https://www.acalculator.org/math/simplifying-radicals-calculator",
  "markdown": "https://www.acalculator.org/math/simplifying-radicals-calculator.md",
  "kind": "function",
  "method": "Factor the radicand into primes, x = kⁿ × m, where kⁿ is its largest perfect nth-power factor; then a × ⁿ√x = (a × k) × ⁿ√m.",
  "assumptions": [
    "The number in front and the radicand are whole numbers; the index is a whole number from 2 to 10.",
    "An odd root of a negative number is negative (∛−8 = −2); an even root of a negative number has no real value.",
    "The root is the principal root: √x is never negative."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "a": {
        "title": "Number in front (a)",
        "description": "The whole number that multiplies the radical, as in 3√50; 1 when there is none.",
        "type": "integer",
        "minimum": -1000000,
        "maximum": 1000000
      },
      "n": {
        "title": "Index (n)",
        "description": "Which root: 2 for a square root, 3 for a cube root, up to 10.",
        "type": "integer",
        "minimum": 2,
        "maximum": 10
      },
      "x": {
        "title": "Radicand (x)",
        "description": "The whole number under the radical sign, up to 1 trillion in size.",
        "type": "integer",
        "minimum": -1000000000000,
        "maximum": 1000000000000
      }
    }
  },
  "outputs": {
    "simplified": {
      "label": "Simplest radical form",
      "description": "The radical with every perfect nth-power factor taken out, such as 6√2.",
      "format": "text"
    },
    "decimal": {
      "label": "Decimal value",
      "description": "The value of the radical as a decimal.",
      "format": "number"
    },
    "outside": {
      "label": "Number outside",
      "description": "The whole number in front of the simplified radical.",
      "format": "integer"
    },
    "inside": {
      "label": "Number inside",
      "description": "The radicand left under the radical sign; 1 when the root is a whole number.",
      "format": "integer"
    },
    "radical": {
      "label": "Radical",
      "description": "The radical as typed, such as 3√50.",
      "format": "text"
    },
    "steps": {
      "label": "Working",
      "description": "The prime factors of the radicand, the largest perfect nth-power factor, and the simplified form.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "a": 1,
      "n": 2,
      "x": 72
    },
    "outputs": {
      "simplified": "6√2",
      "decimal": 8.485281374238571,
      "outside": "6",
      "inside": "2",
      "radical": "√72",
      "steps": "72 = 2^3 × 3^2; The largest perfect square factor is 36 = 6^2, so 72 = 36 × 2; √72 = 6√2"
    },
    "text": "√72 in simplest radical form is 6√2."
  },
  "examples": [
    {
      "given": {
        "a": 1,
        "n": 2,
        "x": 72
      },
      "expect": {
        "simplified": "6√2",
        "outside": 6,
        "inside": 2,
        "decimal": 8.48528137423857,
        "steps": "72 = 2^3 × 3^2; The largest perfect square factor is 36 = 6^2, so 72 = 36 × 2; √72 = 6√2"
      },
      "source": "hand calculation in content.mdx; Python 3: 6 * math.sqrt(2) = 8.48528137423857, math.sqrt(72) = 8.48528137423857; method from OpenStax Elementary Algebra 2e, section 9.2: https://openstax.org/books/elementary-algebra-2e/pages/9-2-simplify-square-roots"
    },
    {
      "given": {
        "a": 3,
        "n": 2,
        "x": 50
      },
      "expect": {
        "simplified": "15√2",
        "outside": 15,
        "inside": 2,
        "radical": "3√50"
      },
      "source": "hand calculation in content.mdx: 50 = 25 × 2, so 3√50 = 3 × 5√2 = 15√2"
    },
    {
      "given": {
        "a": 1,
        "n": 3,
        "x": -16
      },
      "expect": {
        "simplified": "-2∛2",
        "steps": "∛(-16) = -∛16, because an odd root keeps the sign; 16 = 2^4; The largest perfect cube factor is 8 = 2^3, so 16 = 8 × 2; ∛(-16) = -2∛2"
      },
      "source": "hand calculation in content.mdx: 16 = 2³ × 2; Python 3: -2 * 2 ** (1/3) = -2.5198420997897464"
    },
    {
      "given": {
        "a": 1,
        "n": 4,
        "x": 162
      },
      "expect": {
        "simplified": "3∜2",
        "decimal": 3.5676213450081633
      },
      "source": "hand calculation in content.mdx: 162 = 3⁴ × 2; Python 3: 162 ** 0.25 = 3.5676213450081633"
    },
    {
      "given": {
        "a": 1,
        "n": 2,
        "x": 144
      },
      "expect": {
        "simplified": "12",
        "inside": 1,
        "decimal": 12
      },
      "source": "hand calculation in content.mdx: 144 = 12²"
    },
    {
      "given": {
        "a": 2,
        "n": 5,
        "x": 96
      },
      "expect": {
        "simplified": "4 × ⁵√3",
        "outside": 4,
        "inside": 3
      },
      "source": "hand calculation in content.mdx: 96 = 2⁵ × 3, so 2 × ⁵√96 = 2 × 2 × ⁵√3"
    },
    {
      "given": {
        "a": 1,
        "n": 2,
        "x": 30
      },
      "expect": {
        "simplified": "√30",
        "outside": 1,
        "inside": 30
      },
      "source": "hand calculation in content.mdx: 30 = 2 × 3 × 5 has no square factor"
    }
  ],
  "sources": [
    "OpenStax, Elementary Algebra 2e, section 9.2 Simplify Square Roots (product property of square roots): https://openstax.org/books/elementary-algebra-2e/pages/9-2-simplify-square-roots",
    "OpenStax, Intermediate Algebra 2e, section 8.2 Simplify Radical Expressions (higher roots): https://openstax.org/books/intermediate-algebra-2e/pages/8-2-simplify-radical-expressions"
  ],
  "related": [
    "square-root",
    "root",
    "exponent",
    "factor"
  ],
  "changelog": []
}