Square Root Calculator

√n is the number that squares to n: √144 = 12, because 12 × 12 = 144. Squaring and square-rooting undo each other, which is why this calculator shows n² next to the roots — the two columns are the same operation read in opposite directions. Perfect squares — 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 — are the numbers whose roots come out whole. Most numbers are not perfect squares, and their roots are decimals that never terminate and never repeat: √2 ≈ 1.4142135… runs on forever. There is a fast way to rule a number out. A perfect square can only end in 0, 1, 4, 5, 6 or 9. Anything ending in 2, 3, 7 or 8 cannot possibly be one, so 1,048 and 22,222 are out without a single division.

The equation x² = 4 has two solutions, +2 and −2, because squaring destroys the sign. Yet √4 is defined as 2 alone. That looks like an inconsistency and is the single most common confusion around roots. The resolution: the radical symbol √ does not mean "any number that squares to this". It denotes the principal square root, defined to be the non-negative one. It has to be a single value, because √ is a function, and a function is only allowed one output per input. If √4 were both +2 and −2, you could not write it inside a larger expression without ambiguity. This is exactly why the quadratic formula spells out ±√Δ. That ± is not decoration — it is there precisely because √ on its own would only supply the positive root, and both are needed. One consequence catches people out in algebra: √(x²) is not x. It is |x|, the absolute value. Substitute x = −5 and you get √25 = 5, not −5. Cancelling a square against a square root without keeping the absolute value is a standard way to lose a solution or invent a false one.

You can get within a fraction of a percent with nothing but mental arithmetic. Start by bracketing between the perfect squares you know. For √50: 49 is 7² and 64 is 8², so the answer sits just above 7. Then refine with an averaging step that has been used for millennia. Take a guess g and replace it with (g + n/g) ÷ 2. Starting at g = 7 for n = 50: 50 ÷ 7 = 7.142857, and averaging with 7 gives 7.0714286. Run it once more and you get 7.0710678, which is correct to seven decimal places. The method roughly doubles the number of correct digits at each step, and it is the same idea that Newton's method generalises. Why the average works is worth seeing. If your guess is too small, then n/g is too large by a compensating amount, so the true root is trapped between them — and the midpoint is a far better estimate than either. This is also, in essence, what the hardware inside your processor is doing when you press the root key, just with a cleverer starting guess.

The decimal expansion of √2 does not stop and does not fall into a repeating pattern. That is not a limitation of our notation or a matter of not having computed far enough — √2 is irrational, meaning it cannot be written as a fraction of two whole numbers at all. The classical proof is short. Suppose √2 = p/q in lowest terms. Squaring gives p² = 2q², so p² is even, which forces p itself to be even (odd numbers square to odd numbers). Write p = 2k, substitute, and you get 4k² = 2q², so q² = 2k² and q must be even too. But p and q were in lowest terms and cannot both be even. The assumption collapses, so no such fraction exists. The same argument works for the square root of any number that is not a perfect square, which means almost every square root you will ever compute is irrational. Any decimal a calculator gives you is a rounded approximation. When exactness matters, leave the answer as √50 or simplify it to 5√2 rather than committing to 7.0710678. And the number is far from exotic. √2 is the diagonal of a one-unit square, and the reason doubling the area of a room only widens it by about 41%: to double area you multiply each dimension by √2 ≈ 1.414.

A square root is the same thing as raising to the power ½: √n = n^0.5. The cube root ∛n undoes cubing, so ∛27 = 3, and in exponent form it is n^(1/3). Every root is a fractional power, which is why one set of exponent rules covers both. Negative inputs behave differently for the two. A negative number has no real square root, because squaring any real number — positive or negative — produces something positive. Nothing real squares to −9. That gap is what the imaginary unit fills, defined by √−1 = i, giving √−9 = 3i. This calculator works in real numbers, so it reports a negative input as undefined. Cube roots have no such problem. ∛−27 = −3, a perfectly ordinary real number, because multiplying three negatives together leaves the result negative. The general rule: odd roots of negative numbers are real, even roots are not. Software does not always follow that rule cleanly. In JavaScript, Math.pow(-27, 1/3) returns NaN rather than −3, because the exponent is not an integer; Math.cbrt(-27) returns the −3 you wanted. When you need an odd root of a negative number, reach for the dedicated function instead of the power operator.