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Square Root Calculator

Calculate square roots, cube roots and any nth root up to 100. Whole numbers also get the simplified radical form, such as √72 = 6√2, with prime-factor steps.

  • Free, no sign-up
  • Runs in your browser
  • Math
Root
Any number from -1e15 to 1e15. Whole numbers also get the simplified radical form.

How to use the calculator

Enter a number and choose the root: Square, Cube or Other for any whole-number degree from 2 to 100. Press Calculate. The number can be any value from -1e15 to 1e15, including decimals.

The result shows the principal root with up to 10 significant digits. For whole numbers you also get the simplified radical form and whether the number is a perfect square, cube or higher power. When the root can be simplified, the details panel shows the prime-factor steps.

How simplifying a radical works

A square root can be simplified when the number has a factor that is a perfect square. The calculator finds it from the prime factorization:

  1. Break the number into prime factors.
  2. Group equal factors in pairs for a square root, in threes for a cube root, or in groups of n for an nth root.
  3. Each complete group comes out of the root as a single factor.
  4. Whatever is left over stays under the root.

Worked examples

√72. 72 = 2³ × 3². Split 2³ into 2² × 2. The pairs 2² and 3² come out as 2 × 3 = 6, and a single 2 stays under the root, so √72 = 6√2 ≈ 8.485281374.

∛-27. An odd root of a negative number is a real, negative number: (-3)³ = -27, so ∛-27 = -3.

√-16. No real number squared gives a negative result. Using the imaginary unit i, where i² = -1, √-16 = √16 × √-1 = 4i.

Good to know

  • Every positive number has two square roots, for example 3 and -3 for 9. The √ sign means the principal (non-negative) one, and that is the value shown.
  • A whole number that is not a perfect square has an irrational square root: its decimal digits never end or repeat, so the decimal shown is rounded.
  • For even roots above 2 of a negative number, such as the 4th root of -16, there is no real result. The calculator shows the principal complex root instead, which for that example is about 1.414213562 + 1.414213562i.
  • Raising the result to the power n gives back the original number, which is an easy way to check it.

Frequently asked questions

What is the simplified radical form?

It is the root written with the largest possible whole number outside the radical sign and the smallest possible number inside. For example √72 becomes 6√2 because 72 = 36 × 2 and √36 = 6. If the number has no square factor other than 1, like √6, it is already in simplest form.

How do I know if a number is a perfect square?

A whole number is a perfect square when its square root is a whole number, such as 144 = 12². In the prime factorization of a perfect square every prime appears an even number of times. The calculator tells you the answer for any whole number you enter, and does the same for cubes and higher powers.

Can I take the square root of a negative number?

Not as a real number. The result is imaginary: √-16 = 4i, where i is the square root of -1. Odd roots are different: the cube root of a negative number is a real negative number, for example ∛-27 = -3.

Why does the decimal result look rounded?

Most roots are irrational, so their decimal expansion goes on forever. The calculator shows up to 10 significant digits, which is more than enough for everyday use. The simplified radical form, such as 6√2, is the exact value.