Enter any number and this calculator returns its square root as a decimal to your chosen precision and, where one exists, the exact simplified radical form such as 6√2 — the answer most homework and exam questions actually ask for. Switch the root dropdown for cube roots, 4th roots or any custom nth root. It also flags perfect squares, shows the prime factorisation used to simplify, and handles negatives and decimals correctly. See the exponent calculator for roots written as fractional powers, or the standard deviation calculator for a common real-world use.
Square Root Calculator
Square roots as exact simplified radicals, not just decimals — plus cube roots and any nth root.
Try one
Square root
—
Simplified radical—
Perfect power?—
Squared back—
Negative root—
Prime factorisation
Embed this calculator on your site — free
What a square root is
The square root of a number is the value that, multiplied by itself, gives that number. √49 = 7 because 7 × 7 = 49. Strictly every positive number has two square roots — 7 and −7 both square to 49 — but the radical symbol √ refers to the principal (positive) root by convention, which is why calculators return 7 rather than ±7.
Simplified radical form
This is the part most calculators skip. √72 as a decimal is 8.485281…, which is an irrational number that never terminates. But √72 can be written exactly as 6√2, because 72 = 36 × 2 and 36 is a perfect square. That exact form is what maths teachers and exam mark schemes usually want, and it is what this calculator gives you alongside the decimal.
The method: factor the number, pull out any pair of identical prime factors as a single factor outside the radical, and leave the rest inside. For 72 = 2 × 2 × 2 × 3 × 3, the pair of 2s becomes a 2 outside, the pair of 3s becomes a 3 outside, and a single 2 stays inside — giving 2 × 3 × √2 = 6√2.
Perfect squares
A perfect square is a number whose square root is a whole number: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and so on. When you enter one, this calculator says so, and the simplified radical collapses to a plain integer with nothing left under the sign.
Cube roots and nth roots
A cube root asks which number cubed gives your input — ∛27 = 3. Unlike square roots, cube roots handle negatives fine: ∛(−27) = −3, because (−3)³ = −27. The same holds for any odd root. Even roots (square, 4th, 6th…) of a negative number have no real answer, since no real number raised to an even power is negative.
Negative and decimal inputs
Entering a negative number with an even root returns no real result — the answer exists only in the complex numbers, as an imaginary value. Decimals work normally: √0.25 = 0.5. Note that taking the square root of a number between 0 and 1 makes it larger, not smaller, which surprises people the first time they see it.
Where this shows up
Square roots appear all over practical maths — the distance between two points, standard deviation, the quadratic formula, and Pythagoras. If you are working through a quadratic, our quadratic equation calculator handles the whole thing including the discriminant, and the exponent calculator covers roots expressed as fractional powers.
Related calculators
Frequently asked questions
What is the square root of a negative number?
There is no real square root of a negative number, because squaring any real number gives a positive result. The answer exists in the complex numbers as an imaginary value — √(−9) = 3i. Odd roots are different: the cube root of −27 is −27, a perfectly ordinary real number.
How do I write a square root in simplified radical form?
Break the number into factors where one is the largest possible perfect square, then take that square's root outside the sign. √72 = √(36 × 2) = 6√2. This calculator does the factorisation and shows you the working.
Is the square root of 2 rational?
No. √2 is irrational — it cannot be written as a fraction, and its decimal expansion (1.41421356…) never terminates or repeats. The same is true of the square root of any number that is not a perfect square.
Why does √49 give 7 and not −7?
Both 7 and −7 square to 49, but the radical symbol conventionally means the principal (non-negative) root, so √49 = 7. When solving an equation like x² = 49 you must write x = ±7, which is why the negative root is shown separately above.
What is the difference between a square root and a cube root?
A square root asks what number times itself gives your input; a cube root asks what number times itself twice does. √64 = 8 while ∛64 = 4. Cube roots accept negative inputs, square roots do not.
How do I calculate a square root by hand?
The practical method is estimate-and-refine: find the two perfect squares your number sits between, guess a value, then average your guess with the number divided by the guess and repeat. For √50, start at 7 (since 49 is close), then (7 + 50/7)/2 = 7.071 — already accurate to three decimals.