Square Root Calculator

Square root calculator at a glance#

A square root calculator finds the number that, multiplied by itself, gives the number you enter. The square root of x, written √x, is the value y where y times y equals x. So √25 is 5 because 5 times 5 is 25. Negative inputs return an imaginary result.

Those are perfect squares, so their roots are whole numbers. For numbers in between, the root is a decimal. The square root of 50 is about 7.07, because 7.07 times 7.07 is roughly 50. To estimate one by hand, find the two perfect squares it falls between: 50 sits between 49 (root 7) and 64 (root 8), so √50 is just above 7.

Enter any number in the calculator above for the exact square root to several decimal places. Non-perfect-square roots are irrational, so the result is rounded rather than exact.

Finding a square root by hand#

For a number that is not a perfect square, you can close in on its root by averaging. Take √12 as an example.

1. Bracket it: 12 sits between 9 (root 3) and 16 (root 4), so the root is between 3 and 4. Start with the guess 3.5.
2. Average the guess with 12 divided by the guess: (3.5 + 12 ÷ 3.5) ÷ 2 = 3.4643.
3. Repeat with the new value: (3.4643 + 12 ÷ 3.4643) ÷ 2 = 3.4641.

The estimate has already settled, so √12 ≈ 3.4641, which checks out because 3.4641 × 3.4641 ≈ 12. Each round roughly doubles the number of correct digits.

Simplifying a square root#

Pull out any perfect-square factor to write a root in its simplest surd form. For √72, split 72 into 36 × 2, take the root of the perfect square, and leave the rest under the sign: √72 = √36 × √2 = 6√2. The same product rule, √(a × b) = √a × √b, also lets you combine roots: √6 × √2 = √12 = 2√3.

Like terms add the way variables do: 4√2 + 7√2 = 11√2. Unlike roots such as √5 + √3 do not combine further.

Where square roots show up#

The Pythagorean theorem gives the hypotenuse of a right triangle as c = √(a² + b²), so a 3-4 triangle has hypotenuse √(9 + 16) = √25 = 5. The quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a, carries a square root at its core. The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].

Square root FAQ#

What is the square root of a negative number?#

No real number squares to a negative, so the result is imaginary. The square root of −25 is 5i, where i is defined by i² = −1. The calculator above returns this imaginary form for negative inputs.

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

Its square root is a whole number. 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 are the perfect squares up to 100. If the calculator returns a non-terminating decimal, such as √2 ≈ 1.41421, the input is not a perfect square and its root is irrational.

Can this find cube roots and other roots?#

Yes. Switch to the nth-root mode and set n: with n = 3 it returns a cube root, so ∛27 = 3 because 3 × 3 × 3 = 27. Any positive root order works the same way.

Are all square roots irrational?#

No. The roots of perfect squares are rational whole numbers, like √9 = 3. Only the roots of non-perfect squares, such as √2 and √3, are irrational and run on without repeating.

Is √x the same as x to the power one half?#

Yes. The radical form √x and the exponent form x^1/2 denote the same principal square root. The exponent form is handy in algebra because it follows the ordinary power rules.