What a square root really asks
The square root of a number answers a simple question in reverse: which number, multiplied by itself, gives this one? Squaring 12 gives 144, so the square root of 144 is 12. Every positive number has two square roots in principle — 12 and −12 both square to 144 — but by convention the square root symbol and this calculator return the principal root, the positive one. Zero has exactly one root, itself, and negative numbers have no real root at all, because a real number times itself can never be negative: positive times positive is positive, and negative times negative is also positive.
Square roots appear everywhere once you start looking. The side of a square garden with a known area is the square root of that area. The distance between two points on a map uses a square root at the end of the Pythagorean calculation. Screen sizes, falling-object times, standard deviation in statistics and the quadratic formula all end with a root, which is why being able to estimate one — and to simplify one exactly — is such a useful habit.
Perfect squares and simplified radicals
When the root is a whole number, the starting number is a perfect square: 1, 4, 9, 16, 25, 36 and so on. Most numbers are not perfect squares, and their roots are messy decimals — but the exact form can often be tidied. Find the largest perfect-square factor of the number, take its root, and leave the rest under the sign. So √72 becomes 6√2, because 72 = 36 × 2 and √36 = 6. That simplified form is exact, easier to read than 8.485281…, and is the form teachers and textbooks expect.
How to estimate a root by hand
You can pin down any root between two whole numbers without a calculator. Because 8 squared is 64 and 9 squared is 81, the root of 72 must lie between 8 and 9 — and since 72 is closer to 64 than to 81 in one sense but the squares spread out as they grow, a good first guess is about 8.5, whose square is 72.25, remarkably close. One refinement step — average your guess with the number divided by your guess — is the ancient Babylonian method, and it converges astonishingly fast: (8.5 + 72 ÷ 8.5) ÷ 2 is about 8.4853, already correct to four decimal places. The table below the calculator, showing squares and roots from 1 to 20, is worth learning by heart up to 15 squared = 225, because those landmarks make every estimate faster.
Reading the results on this page
The headline gives the root as a decimal. The breakdown also reports whether your number is a perfect square, the simplified radical where one exists, and x squared — the inverse operation — so you can check the answer by squaring it back. For a whole-number input the simplified radical is always shown; for a decimal input such as 2.5 there is no whole-number factorisation, so the page honestly shows the decimal root alone rather than inventing a tidy form that does not exist. Try a few favourites: √2 is about 1.4142, √3 is about 1.7321, and √10 is about 3.1623 — three irrational roots that turn up constantly in geometry.
Worked example
The square root of 144 is 12, because 12 × 12 = 144, so 144 is a perfect square. The square root of 72 simplifies: 72 = 36 × 2, so √72 = 6√2, which is about 8.4853 as a decimal.
How to use this calculator
- Enter the number you want the square root of.
- Read the root in the green panel — a whole number means a perfect square.
- Check the simplified radical for the exact tidy form, if one exists.
- Use x squared in the breakdown to verify: squaring the root should return your number.
- Browse the table of squares and roots from 1 to 20 to build estimation landmarks.
Square roots, cube roots and fractional powers
A square root is one member of a family. The cube root asks which number cubed gives the starting number, and in the language of powers every root is a fractional power: the square root of x is x to the power one half, and the cube root is x to the power one third. That is why the Scientific Calculator can find any root as a power, and why roots and powers undo each other in either order. Once that connection clicks, questions such as the fourth root of 81 stop being exotic — 81 to the power one quarter is 3, because 3 to the fourth power is 81 — and the root, exponent and logarithm tools in this section all start to feel like three views of one idea.