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Root Calculator

Find square, cube and any n-th root, with the exact simplified radical form for whole numbers.

By Nitin Raturi · Last reviewed

Your root

How to use the Root Calculator

Find any root of a number, with the simplified radical form for whole numbers.

  1. Enter the number under the root.
  2. Enter the root: 2 for square, 3 for cube.
  3. Read the decimal root and the simplified radical.

Good to know

  • A simplified radical keeps the exact value; the decimal is rounded.
  • For a fractional power such as x^1.5, use the Exponent Calculator.

Whole numbers also get a simplified radical form.

Number

The number under the root sign (the radicand).

How to fill it in

  • Enter any number; a whole number also gets a simplified radical form.
  • Negative numbers work for odd roots only.

Example

√72 = 6√2 ≈ 8.485281, and √144 = 12.

Root (index)

Which root to take: 2 for a square root, 3 for a cube root, and so on.

How to fill it in

  • Enter a whole number from 2 to 100.

Example

The cube root of −54 is −3∛2 ≈ −3.779763.

Keep in mind: An even root of a negative number is not a real number.

√72

8.485281

√72 simplifies to 6√2, which is about 8.485281.

Simplified radical
6√2
Perfect power?
No

Real roots only. The decimal is rounded to six places; the radical form is exact.

ChangelogHow we test this calculatorEmbed this calculator

How this calculator works

  1. Step 1Factor72 = 2³ × 3²Split the number into prime powers.
  2. Step 2Group(2 × 3)² × 2Take out every group of n equal factors.
  3. Step 3Simplify6√2Groups come outside; the rest stays inside.

The n-th root of x is the number that gives x when raised to the power n. The calculator gives the decimal and, for whole numbers, the exact simplified radical.

Odd roots of negative numbers are negative. Even roots of negative numbers have no real answer, so the calculator says so.

Worked example: the square root of 72

√72 in simplest radical form
Product rule
√(a² × b) = a√b
  • xNumber 72
  • nRoot 2
  1. Prime factors72 = 2 × 2 × 2 × 3 × 32³ × 3²
  2. Pairs out2² × 3² = 36 = 6²6 outside
  3. Left inside72 ÷ 362
  4. Answer6√2 ≈ 6 × 1.4142146√2 ≈ 8.485281

The radical form is exact; the decimal is rounded. Use the radical in algebra and the decimal for measurement.

Three kinds of answer

  • √14412A perfect square: the root is whole.
  • ∛−54−3∛2An odd root keeps the minus sign: about −3.779763.
  • √12.253.5A decimal gets a decimal root only.

Read the result

If the simplified radical has no root sign left, the number is a perfect power and the root is exact.

The decimal root is shown to six places. It is accurate to about 15 significant digits before rounding.

Assumptions and limits

  • Real rootsNo complex answers.
  • Even root of a negativeNot a real number.
  • Whole indexFrom 2 to 100.
  • Radical formFor whole numbers up to 1,00,00,00,00,000.

Simplifying a root means finding square factors. Powers and prime factors are the tools behind it.

To see a number’s prime factors, use the prime factorization calculator. The exponent calculator handles fractional powers, which are roots written another way.

Common mistakes

  • Mistake: Writing √(a + b) = √a + √b.

    Instead: Only products split: √(ab) = √a × √b.

  • Mistake: Leaving √72 as 2√18.

    Instead: Take out the largest square: 6√2.

  • Mistake: Taking √(−4) as −2.

    Instead: (−2)² is 4, not −4; there is no real square root.

Checking the method

Every golden case is re-worked by bisection to 50 digits, and each radical by trying every whole factor.

An independent script finds roots without any power or root function, and simplifies radicals without prime factorisation.

Read how we test our calculators, or browse more maths calculators.

Frequently asked questions

How do I simplify a square root?

Take out the largest perfect square factor: √72 = √(36 × 2) = 6√2.

What is the square root of 144?

12, because 12 × 12 = 144. It is a perfect square.

Can I take the cube root of a negative number?

Yes. Odd roots of negative numbers are negative: ∛−54 = −3∛2 ≈ −3.779763.

Why is there no square root of a negative number?

Any real number squared is zero or positive, so no real number squares to a negative.

What is an nth root?

The number that gives x when multiplied by itself n times. The 5th root of 32 is 2.

Sources and methodology

The formulas are standard time-value-of-money results. We check every figure against golden values worked out independently, by closed form and by month-by-month simulation; see the methodology.

Written and maintained by Nitin Raturi, a software engineer (about the author). Last reviewed 3 Nov 2026; next review due by . Found an error? Tell us.

Changelog

  • : Initial release.