Square Root Calculator
Square root (√), cube root and any nth roots calculated online
- Updated
2 = square root, 3 = cube root
√144
12
12The square root of 144 is 12.
How this was calculated
The nth root is the number that, multiplied by itself n times, gives back the original number. Put simply: the square root of an area is the side of the matching square.
RZ-26 · headon.pro
Roots: basics and formulas
Square root, cube root and nth root explained simply
The square root is one of the fundamental mathematical operations. The square root √x finds the positive number that, multiplied by itself, gives x. This operation appears everywhere: in the Pythagorean theorem (c = √(a² + b²)), in the standard deviation in statistics, in physics formulas, and when calculating side lengths of geometric figures from their areas.
The nth root generalizes the square root: ⁿ√x = x^(1/n). The cube root (degree 3), for example, is needed to determine the edge length of a cube from its volume: if V = a³, then a = ³√V. From a cube with volume 27 cm³, the edge length follows as ³√27 = 3 cm. Fourth roots (degree 4) play a role in thermodynamics (Stefan–Boltzmann law) and in signal processing.
An important restriction: in the real number space, even roots (degree 2, 4, 6 …) cannot be taken from negative numbers. The solution would be a complex number, which is rarely needed in everyday life. For odd degrees (3, 5, 7 …), however, the root of a negative number is also real: ³√(−27) = −3, because (−3)³ = −27. Our calculator gives a clear error message for invalid inputs.
For students, root calculations are a central topic in lower and middle school. Particularly frequently asked: √144 = 12 (reference value for 12 × 12 = 144), √169 = 13, √225 = 15. These values should be known by heart, since they regularly appear in exams and everyday life (e.g. area calculations). With our calculator, you can quickly check not only integer but also fractional radicands and root degrees.
Calculation examples
- Radicand
- 144
- Root degree
- 2 (square root)
- Formula
- 144^(1/2)
- Result
- 12
- Radicand
- 27
- Root degree
- 3 (cube root)
- Formula
- 27^(1/3)
- Result
- 3
Frequently asked questions about the square root calculator
Formulas, examples and tips for calculating roots
How do you calculate a square root?
The square root of a number x (written √x or x^½) is the number that, multiplied by itself, gives x. √144 = 12, because 12 × 12 = 144. For non-integer results (e.g. √2 ≈ 1.41421) our calculator outputs a decimal number with up to 6 digits. Technically, JavaScript calculates √x using Math.sqrt(x).
What is an nth root?
The nth root of a number x is the value y for which: yⁿ = x. The square root (n=2) is the special case. The third root (cube root, n=3) of 27 is 3, because 3³ = 27. In general: ⁿ√x = x^(1/n). In our calculator, you can freely choose the degree n.
Can you take the root of a negative number?
For even root degrees (2, 4, 6, …), roots of negative numbers are not defined in the real number space. √(−4) does not yield a real number, since no real number multiplied by itself gives −4. For odd root degrees (3, 5, 7, …), however, there is a real solution: ³√(−8) = −2, because (−2)³ = −8. Our calculator handles both cases correctly.
What is the difference between a square root and a cube root?
The square root (degree 2) finds the number that, squared, gives the radicand: √x = x^(1/2). The cube root (degree 3) finds the number that, cubed, gives the radicand: ³√x = x^(1/3). Typical use cases: square roots in the Pythagorean theorem and for areas, cube roots for volume calculations (e.g. the edge length of a cube from its volume).
How are roots and powers related?
Taking a root and raising to a power are inverse operations: the nth root of x corresponds to x to the power of 1/n. Therefore ⁿ√(xⁿ) = x holds for positive numbers. This relationship is important in algebra and analysis: it allows roots to be rewritten using power rules. Our square root calculator internally uses Math.pow(x, 1/n), which corresponds exactly to this formula.