Calculate square roots, cube roots, and any nth root. Shows decimal result, simplified radical form, and verifies the answer. Perfect for students and engineers.
The root of a number is the inverse operation of exponentiation. While 3^2 = 9 asks "what is 3 squared?", β9 = 3 asks "what number, squared, gives 9?" Roots are essential in geometry, physics, engineering, statistics, and finance β from the Pythagorean theorem to the quadratic formula to calculating standard deviation.
The square root of a number x (written βx or x^(1/2)) is the value that, when multiplied by itself, equals x. The square root of 25 is 5 because 5 Γ 5 = 25. Square roots have two solutions: both 5 and β5 squared give 25. In practice, the positive root (principal square root) is used by default.
| n | nΒ² | β(nΒ²) | Perfect Square? |
|---|---|---|---|
| 1 | 1 | 1 | Yes |
| 2 | 4 | 2 | Yes |
| 3 | 9 | 3 | Yes |
| 4 | 16 | 4 | Yes |
| 5 | 25 | 5 | Yes |
| 6 | 36 | 6 | Yes |
| 7 | 49 | 7 | Yes |
| 8 | 64 | 8 | Yes |
| 9 | 81 | 9 | Yes |
| 10 | 100 | 10 | Yes |
Many square roots are irrational numbers β they cannot be expressed as a fraction and their decimal representation never terminates or repeats. β2 β 1.41421356..., β3 β 1.73205..., β5 β 2.23607.... These are written in simplified radical form by factoring out perfect squares from under the radical sign.
Method: To simplify β72, factor 72 as 36 Γ 2. Since 36 is a perfect square: β72 = β(36 Γ 2) = β36 Γ β2 = 6β2.
This calculator shows simplified radical form wherever possible.
The cube root of x (βx or x^(1/3)) is the value that, when multiplied by itself three times, equals x. β27 = 3 because 3 Γ 3 Γ 3 = 27. Unlike square roots, cube roots can be taken of negative numbers: β(β8) = β2 because (β2)^3 = β8.
The general nth root of x (written βΏβx or x^(1/n)) is the value y such that y^n = x. Fourth roots, fifth roots, and higher are less commonly used but appear in advanced mathematics, signal processing, and engineering calculations.
Most square roots of non-perfect-square integers are irrational. This was first proved by the ancient Greeks for β2. Irrational numbers include Ο, e (Euler's number), and most roots. They are real numbers but cannot be expressed as fractions. This is why simplified radical form (like 6β2 instead of 8.485...) is mathematically preferable for exact answers.
In a right triangle with legs a and b and hypotenuse c: c = β(aΒ² + bΒ²). If a = 3 and b = 4, then c = β(9 + 16) = β25 = 5. This is the most used square root calculation in construction, carpentry, and engineering throughout both the USA and UK.
The quadratic formula x = (βb Β± β(bΒ² β 4ac)) / 2a uses a square root to solve any quadratic equation. The expression under the square root (bΒ² β 4ac) is called the discriminant β if it is negative, the equation has no real solutions (studied in UK A-Level and US precalculus).
RMS voltage (used in AC electrical systems) is calculated using a square root. The RMS of a sinusoidal AC wave is the peak voltage divided by β2. In the UK, mains electricity is 230V RMS (with a peak of approximately 325V). In the US, it is 120V RMS (peak ~170V). The formula Vrms = Vpeak / β2 directly uses the square root.
Standard deviation is the square root of variance: Ο = β(Ξ£(xβΞΌ)Β²/n). This is used in finance to measure investment volatility, in science to express measurement uncertainty, and in manufacturing quality control. The square root is what converts variance (which is in squared units) back to the original units.
β2 β 1.41421356... It is an irrational number β its decimal representation never terminates or repeats. It appears in the diagonal of a unit square (by the Pythagorean theorem: β(1Β²+1Β²) = β2).
A perfect square is an integer whose square root is also an integer. The perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... The square root of any non-perfect-square integer is irrational.
Factor out all perfect square factors from under the radical. Example: β72 = β(36Γ2) = 6β2. The calculator does this automatically.
Not within the real numbers. β(β1) = i, an imaginary number. However, cube roots of negative numbers are real: β(β8) = β2.
Root Mean Square (RMS) is the square root of the mean of squared values. It's used in electrical engineering to express the effective value of AC voltage/current. UK mains is 230V RMS, US mains is 120V RMS.
β27 = 3, because 3Β³ = 3 Γ 3 Γ 3 = 27. Other perfect cubes: β8=2, β64=4, β125=5, β216=6, β1000=10.
The quadratic formula x = (βb Β± β(bΒ²β4ac))/2a uses the square root of the discriminant. If bΒ²β4ac > 0 there are two real roots; if = 0 one root; if < 0 no real roots.
The nth root of x is the value y such that y^n = x. Written as x^(1/n) or βΏβx. For n=2 it's the square root, n=3 the cube root, etc. This calculator handles any positive integer n.
Results are for educational purposes. Floating-point arithmetic may introduce small rounding errors for very large numbers. Always verify safety-critical engineering calculations with certified tools.