TheCalculatory
Math · Everyday Arithmetic Formula verified

Square Root Calculator

Enter a number to find its square root, or switch to nth root for any other degree. Integer square roots are also shown in simplified radical form.

Root
05001000
On-screen number pad
Tap to type into the highlighted field
Square root
8.485281
√72
Simplified radical form
6√2
Perfect square?
No

Step-by-step proofCheck by hand

How each root is calculated

The square root of a number is whatever value, multiplied by itself, gives that number back. For an integer, that root can often be simplified: if the number has a perfect-square factor (4, 9, 16, 25, and so on), that factor's own root can be pulled out in front of the radical sign, leaving a smaller, square-free number underneath. An nth root generalizes the same idea to any degree — a cube root (n = 3) asks what value, cubed, gives the number back.

Negative numbers have no real square root, since squaring anything real gives zero or a positive result. Odd-degree roots of negative numbers do exist and are real, because an odd number of negative factors multiplied together stays negative.

The formulas

√n = n^(1/2)  ·  ⁿ√x = x^(1/n)  ·  √(a²b) = a√b

Worked example

√72: 72 factors as 36 × 2, and 36 is a perfect square (6²), so √72 = √36 × √2 = 6√2, which is about 8.485281 as a decimal.

Frequently asked questions

What does 'simplified radical form' mean?
It means pulling the largest perfect-square factor out from under the root sign. √72 isn't in simplest form as written, because 72 has a perfect-square factor of 36 (72 = 36 × 2); pulling it out gives 6√2, the simplified form. A number with no perfect-square factor greater than 1, like 2, is already fully simplified.
Why is there no real square root of a negative number?
Because any real number squared is zero or positive — a negative times itself is positive, and a positive times itself is positive — so no real number can square to a negative result. Negative numbers do have square roots among the complex numbers, but this calculator sticks to real numbers, which is what almost every practical use calls for.
Why does an odd-degree root of a negative number exist, but not an even one?
An odd number of negative factors multiplied together stays negative, so a negative number cubed, for instance, is still negative — meaning a negative number does have a real cube root. An even number of negative factors always turns positive, so no real number raised to an even power can land back on a negative result.
Sources and method
  • Standard radical arithmetic and simplification (extracting perfect-square factors) — general published mathematics.
Last reviewed: 19 Sep 2026 Sources last verified: 19 Sep 2026 Results use the assumptions explained on this page. Report an error How we check calculations

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