How to use the square root calculator
- Enter a number. Whole numbers up to 1,000,000,000,000 and decimals with up to 3 places get an exact radical form.
- Read the square root to 12 significant digits and the exact form, such as 6√2.
- For whole numbers, check whether the number is a perfect square and which perfect squares it falls between.
- Follow the steps to see the prime factors that come out of the root.
Formula
n = (outside)^2 x inside, where inside has no square factor
square root of n = outside x square root of inside
a decimal p/q in lowest terms: square root = square root of (p x q) / q
negative n: square root = i x square root of |n|
Prime factors that appear an even number of times come out of the root in pairs. Factors left over stay inside.
Simplifying the square root of 72
72 = 2 x 2 x 2 x 3 x 3. The pair of 2s and the pair of 3s come out as 2 x 3 = 6, and one 2 stays inside, so the square root of 72 is 6√2, about 8.48528137424. 72 is between the perfect squares 64 and 81.
The square root of 12.5 is 25/2 under the root, which is √50 / 2 = 5√2 / 2, about 3.5355. The square root of 0.04 is exactly 1/5. The square root of 144 is 12, a perfect square, and the root of -9 is i x 3.
Simplest radical form
A radical is in simplest form when no perfect square is left inside the root. Factor the number into primes, take out each pair, and multiply what comes out. For example 360 = 2^3 x 3^2 x 5, so the root is 6√10. Teachers often ask for this form because it is exact, while a decimal is rounded.
Limits of the exact form
The page factors by trial division, so exact forms are limited to whole numbers up to 10^12 and decimals with up to three places and a value of at most 1,000,000. Larger or longer numbers still get the decimal root to 12 digits. A number with a huge prime factor has no square factor to pull out, so its root stays under the radical.
Related math tools
Raise numbers to powers and find other roots with the Log, Exponent and Root Calculator, find prime factors with the Prime Number Calculator, work with fractions in the Fraction Calculator, and use roots in the Triangle Calculator.
Assumptions and limits
- The square root shown is the principal (non-negative) root.
- Decimal results are shown to 12 significant digits.
- Exact forms need whole numbers up to 10^12 or short decimals.
- Negative numbers give an imaginary root written with i.
Frequently asked questions
How do you simplify a square root?
Factor the number into primes, move each pair of equal primes outside the root as a single factor and leave the rest inside. The square root of 72 is 6√2.
How do I know if a number is a perfect square?
Its square root is a whole number. The page checks this for whole numbers and shows the two perfect squares around a number that is not one.
What is the square root of a negative number?
It is imaginary. The square root of -9 is i x 3, where i times i equals -1. The page shows this for negative inputs.
What is the square root of a decimal such as 12.5?
Write it as a fraction, 25/2, and take the root of the top times the bottom over the bottom. That gives 5√2 / 2, about 3.5355.