Calculate the square root of any non-negative number with principal and negative roots, equation checks, and nearby perfect-square references.
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Algebra
Calculate square roots with exact-result and nearby perfect-square checks
Enter any non-negative number to find the principal square root, the matching negative root, and whether the result is exact or only a decimal approximation.
Integer inputs also show the simplest exact radical form when a square factor can be pulled out cleanly.
Math note
The square-root symbol refers to the principal positive root. Positive numbers also have a matching negative root when you solve the equation x² = value.
Enter a value Type a non-negative number above to calculate its square root.
Square root calculator with exact radical form, perfect-square detection, and nearby
The square root calculator finds the square root of any non-negative number. It shows both the principal (positive) root and the negative root, identifies perfect squares, flags whether the result is exact or an approximation, and simplifies integer radicals such as √72 into 6√2 when a square factor can be pulled out exactly.
How square roots are calculated
The square root of a number x is the value y such that y × y = x. Every positive number has two square roots: a positive one (the principal root) and a negative one. By convention, the square root symbol refers to the principal root.
Perfect squares — 1, 4, 9, 16, 25, 36, and so on — have integer square roots. Non-perfect squares like 2, 3, and 5 have irrational roots that cannot be expressed as exact fractions.
Some non-perfect-square integers still have a cleaner exact radical form. For example, 72 is not a perfect square, but because 72 = 36 × 2, √72 simplifies to 6√2. That exact form is often more useful in algebra than a decimal approximation.
√x = x^(1/2)
The square root of x is x raised to the power one-half.
√(a² × b) = a√b
Pull square factors outside the radical to simplify exact integer roots.
Why nearby perfect squares matter
Nearby perfect squares tell you instantly where an irrational root must land. If your number sits between 49 and 64, its square root must sit between 7 and 8.
The add or subtract amounts are also practical. If you need an exact square for a geometry problem, a layout problem, or a teaching example, the calculator shows how far your input is from the next lower or upper perfect square.
How to read the exact radical form
When the calculator shows a result like 5√2, it means the root is exact even though it is not an integer. The decimal is only an approximation of that exact form.
This is useful in algebra, trigonometry, and geometry because exact radicals preserve precision. A decimal such as 7.0711 is fine for measurement, but 5√2 is better when you still need to simplify later expressions.
Limitations
Square roots of negative numbers are not real. This calculator handles non-negative inputs only.
The page simplifies radicals for integer inputs only. Decimal inputs can still have exact roots, but the display does not attempt symbolic fraction-based simplification.
Further reading
Britannica — Square root — Reference overview of square roots, irrational values, and the mathematical meaning of root operations.
The square root of 50 is about 7.071. It is not exact because 50 is not a perfect square. The nearest perfect squares are 49 and 64, which come from 7² and 8².
That means √50 sits between 7 and 8. This kind of nearby-square check is useful when you want to estimate the result before looking at the decimal approximation.
The exact radical form is 5√2 because 50 = 25 × 2. That is why the decimal starts a little above 7 instead of much closer to 8.
Frequently asked questions
Is the square root of 2 rational or irrational?
Irrational. It cannot be expressed as a fraction and its decimal expansion (1.41421356...) never terminates or repeats.
Why does every positive number have two square roots?
Because both a positive and a negative number, when squared, give the same positive result. For example, both 3 and -3 squared give 9.
How can I estimate a square root without a calculator?
Find the nearest perfect squares above and below the number. For example, 49 and 64 surround 50, so √50 must be between 7 and 8. The closer the number is to one of those perfect squares, the closer the root is to that whole number.
Why does the calculator show an exact radical like 6√2 for some numbers?
Because some integers contain a square factor even when they are not perfect squares. For 72, the square factor 36 can be pulled out of the radical, so √72 simplifies exactly to 6√2.
Why does the calculator show both positive and negative values?
Because both numbers square to the same positive result. For example, 5² and (-5)² both equal 25. The square-root symbol still refers to the principal positive root by convention.
What does it mean when the calculator says add 14 to reach the next exact square?
It means your current input is 14 below the next perfect square. For example, 50 is 14 below 64, so adding 14 would move the value to 8² and give an exact integer root.
Can a decimal have an exact square root?
Yes. For example, √2.25 = 1.5 exactly. A number does not need to be an integer perfect square to have an exact terminating root.
Can I take the square root of a negative number here?
No. This calculator is limited to real-number square roots, so negative inputs are not evaluated. Negative numbers require complex numbers rather than ordinary real roots.
