Enter a number to get its square root as a decimal and, where possible, the simplified radical form. The factoring steps are shown below the result.
Try a perfect square like 144, or a number with a square factor like 72.
The square root of a number n is the value that, multiplied by itself, equals n. The radical symbol is the check-mark shape placed over the number being rooted.
When n is a perfect square the root is a whole number: sqrt(144) = 12 because 12 x 12 = 144. When n is not a perfect square, the root is irrational and never terminates, so the calculator returns a decimal rounded to several places and, whenever a perfect-square factor exists, the simplified radical form as well.
72 is not a perfect square, but it has a perfect-square factor hiding inside it. Break it down step by step.
A frequent error is stopping at the first perfect-square factor found instead of the largest one; 72 = 4 x 18 also works, but simplifying from 4 leaves sqrt(18) still unsimplified underneath, so you have to repeat the process. It also helps to remember that a negative number has no real square root at all, only an imaginary one written with i, so sqrt(-9) is not "no answer" but rather 3i. Finally, a decimal input like 6.25 can still have an exact root (2.5), so it is worth checking whether the number is a perfect square before assuming the decimal answer is the final word.
For background on radicals and irrational roots, see Wolfram MathWorld's page on the square root.
A square root often shows up right after solving a quadratic or checking a right triangle.
Find a number that multiplies by itself to give your value. For perfect squares like 144 the answer is exact (12). For others, estimate between the two nearest perfect squares, then refine.
It pulls the largest perfect-square factor out of the radical. For example sqrt(72) = sqrt(36 x 2) = 6 sqrt(2).
Not within the real numbers. The square root of a negative number is imaginary, written with i, such as sqrt(-9) = 3i.
Yes. This tool accepts decimals and returns the decimal root rounded to several places.
Marcus Vance spent a decade building data pipelines and testing calculation logic before he started writing about the math behind the tools. On MathCalcTools he checks every formula against the source math, then writes out the steps so a reader can follow the same path a calculator takes.