Find the square root of any number — as a decimal and, where possible, in simplified radical form such as 2√2. Enter a number and the calculator returns its principal square root, both roots (±), the simplified radical, and the number squared. Negative numbers return an imaginary result.
The square root calculator finds the square root of any number instantly. It gives you the answer as a precise decimal, shows both the positive and negative roots, and — when the number allows it — simplifies the square root into its cleanest radical form, like turning √8 into 2√2. It even handles negative numbers by returning an imaginary result. Whether you are checking homework, working through algebra, or just curious what the square root of a number is, this tool gives a complete, accurate answer in one step.
This page explains what a square root is, how to find one, how to simplify radicals, how negative numbers work, and answers the questions people ask most. By the end you will understand not just the answer the calculator gives, but exactly how and why it works.
The square root of a number x is the value that, when multiplied by itself, gives x. In symbols, if y × y = x, then y is a square root of x, written √x. For example, the square root of 25 is 5, because 5 × 5 = 25. Squaring and taking a square root are inverse operations: one undoes the other. The little check mark symbol √ is called the radical sign, and the number under it — the one you are taking the root of — is called the radicand. Square roots are one of the most fundamental operations in mathematics, appearing in geometry, algebra, physics, statistics, and everyday measurement.
A key fact that surprises many people is that every positive number has two square roots — one positive and one negative. For instance, both 5 and −5 are square roots of 25, because 5 × 5 = 25 and (−5) × (−5) = 25 as well (a negative times a negative is positive). The positive one is called the principal square root and is what the radical symbol √ usually refers to, so √25 = 5. The calculator shows both roots — the principal (positive) root and its negative counterpart — because both are valid solutions to the equation y² = x, which matters when you are solving equations rather than just evaluating a root. The number zero is the only number with a single square root: √0 = 0.
For perfect squares — numbers like 1, 4, 9, 16, 25, 36 that are the square of a whole number — the square root is simply that whole number, and you can often recognize it instantly. For all other numbers, the square root is an irrational number with an infinite, non-repeating decimal, so it can only be approximated. Historically people used long-division-style algorithms or lookup tables; today a calculator uses fast numerical methods (like Newton's method) to find the root to many digits in an instant. This calculator returns the decimal to high precision, which is more than enough for schoolwork, construction, or engineering. If you want to estimate a root by hand, find the two nearest perfect squares: √52, for example, lies between √49 = 7 and √64 = 8, and closer to 7, so a good estimate is about 7.2 — and the calculator confirms 7.2111.
Take the number 52. It is not a perfect square, so its square root is an irrational decimal:
√52 ≈ 7.2111 (and −7.2111 is the other root)
But 52 can be simplified in radical form. Since 52 = 4 × 13 and 4 is a perfect square, we can pull the 4 out as a 2:
√52 = √(4 × 13) = √4 × √13 = 2√13
So the square root of 52 is exactly 2√13, approximately 7.2111. These are the calculator's default values, so you can press Calculate to see the decimal, both roots, and the simplified radical, then try your own number.
To simplify a square root means to write it in the form a√b, where a and b are whole numbers and b is as small as possible. The method is to find the largest perfect-square factor of the number under the root, take its square root outside the radical, and leave the rest inside. For √72, the largest perfect-square factor is 36 (since 72 = 36 × 2), so √72 = √36 × √2 = 6√2. If the number under the root has no perfect-square factor other than 1 — like 13, 7, or 15 — then it is already in simplest form and cannot be reduced. Simplified radical form is exact, unlike a rounded decimal, which is why it is preferred in algebra and higher mathematics. This calculator automatically finds the simplest radical form for you whenever the number is a whole number.
Within the ordinary (real) number system, you cannot take the square root of a negative number, because no real number multiplied by itself gives a negative result — a positive times a positive is positive, and a negative times a negative is also positive. To handle these cases, mathematicians defined the imaginary unit i, where i² = −1, so that √(−1) = i. The square root of any negative number is then that of its absolute value multiplied by i: for example, √(−9) = √9 × i = 3i, and √(−20) = 2√5 i. These are called imaginary numbers, and together with real numbers they form the complex numbers used throughout advanced mathematics, engineering, and physics. The calculator returns the imaginary form when you enter a negative number.
Square roots follow a set of handy rules that make working with them easier. The product rule says √(a×b) = √a × √b, which is exactly what powers radical simplification. The quotient rule says √(a/b) = √a / √b, useful for square roots of fractions. Squaring a square root returns the original number: (√x)² = x. A square root is the same as raising to the power one-half: √x = x^(1/2), which connects roots to exponents and explains why calculators handle both with the same machinery. Note, however, that square roots do not distribute over addition: √(a+b) is not equal to √a + √b — a very common mistake. Keeping these properties in mind helps you manipulate and check square-root expressions confidently.
Square roots appear far beyond the math classroom. In geometry, the Pythagorean theorem uses a square root to find the length of a triangle's hypotenuse, and the distance between two points on a plane is a square root of summed squares. In statistics, the standard deviation is the square root of the variance. In physics, square roots appear in formulas for the period of a pendulum, the speed of a wave, and kinetic energy relationships. Finance uses them in volatility and risk calculations, and computer graphics use them constantly to normalize vectors and compute distances. Even everyday tasks — figuring out the side of a square room from its area, or sizing a square TV from its diagonal — rely on square roots. Because the operation is so widespread, a fast, accurate square root calculator is a genuinely useful everyday tool.
Squaring and taking a square root are inverse operations, and it helps to see them side by side. Squaring a number means multiplying it by itself: 6 squared is 6 × 6 = 36. Taking the square root reverses that: the square root of 36 is 6. This calculator shows both — the square root of the number you enter and the number squared — so you can move in either direction. Squaring always produces a non-negative result, which is precisely why negative numbers have no real square root. And because squaring is the inverse of the square root, you can always check a square-root answer by squaring it: if √52 ≈ 7.2111, then 7.2111² should be about 52 — a quick way to confirm any result.
Recognizing perfect squares on sight makes square-root work far faster, both for estimating roots and for simplifying radicals. The perfect squares of the numbers 1 through 15 are worth committing to memory: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100, 11²=121, 12²=144, 13²=169, 14²=196, and 15²=225. Once you know these, the square roots of 1, 4, 9, 16, 25, and so on are instant, and you can bracket any other root between two of them. For example, since 50 sits between 49 (7²) and 64 (8²), you immediately know √50 is a little more than 7 — and closer to 7 than 8, because 50 is much nearer 49. Knowing perfect squares also speeds up radical simplification, because you can quickly spot when a perfect square divides evenly into your number. The calculator does all of this for you, but the mental picture helps you sanity-check its answers and estimate on the fly.
A few errors come up again and again when people work with square roots by hand. The first is assuming √(a + b) equals √a + √b — it does not; for example √(9 + 16) = √25 = 5, not √9 + √16 = 3 + 4 = 7. The second is forgetting the negative root: when you solve an equation like x² = 49, the answer is x = ±7, not just 7, even though the √ symbol alone gives only the positive root. The third is trying to take the square root of a negative number in the real numbers and getting confused — the answer is imaginary, not "error" or "undefined." The fourth is leaving a radical unsimplified when a cleaner form exists, such as writing √50 instead of 5√2. Finally, people sometimes round too early: keep full precision through a calculation and round only the final answer. This calculator avoids all of these pitfalls by showing both roots, simplifying automatically, handling negatives with the imaginary unit, and carrying high precision throughout.
When a square root appears inside a larger expression, it behaves like a grouping symbol: you evaluate everything under the radical first, then take the root, and only then combine it with the rest of the expression. For instance, in √(16 + 9) you add first to get √25 = 5, whereas √16 + 9 means take the root of 16 and then add 9, giving 4 + 9 = 13. The radical bar itself groups its contents, much like parentheses. In the standard order of operations, a square root ranks with exponents, since a root is just a fractional power. Being careful about what is and is not under the radical is one of the most important habits in algebra, and it is exactly why writing expressions clearly matters. When you use this calculator, enter the single value you want the root of; if your problem has an expression under the radical, simplify that expression first and enter the result.
It is the value that, multiplied by itself, gives the original number. For example, the square root of 49 is 7 because 7 × 7 = 49.
Every positive number has two: a positive (principal) root and a negative one, such as +7 and −7 for 49. Zero has one root (0), and negative numbers have imaginary roots.
Find the largest perfect-square factor of the number, take its root outside the radical, and leave the rest inside. For example, √72 = √(36×2) = 6√2.
It is imaginary. Take the square root of the absolute value and multiply by i, where i² = −1. So √(−9) = 3i.
No. √2 ≈ 1.41421 is irrational — its decimal never ends or repeats. Only perfect squares have whole-number square roots.
Square it. If your answer for √x, when multiplied by itself, gives back x, the answer is correct.
This Square Root Calculator is provided for general educational purposes. It returns high-precision decimal results and exact simplified radicals for whole numbers. Because computers store decimals in binary, extremely large inputs may show tiny rounding differences in the last digits.