Math · Live
Root Calculator:
square, cube, and nth root.
Calculate the square root, cube root, or any nth root of any number. Supports negative radicands with odd roots, detects perfect roots, shows step-by-step working, and compares all common roots and powers side by side.
Inputs
Radicand and degree
Decimal places
- ²√2
- 1.4142135624
- Type
- Irrational (real)
- Fractional exp
- x^(1/2)
Square root
²√2
²√2 = 2^(1/2) = 2^{0.500000}
Working
Step-by-step
- 1Fractional exponent identity
ⁿ√x = x^(1/n)
- 2Substitute values
²√2 = 2^(1/2) = 2^(0.50000000)
- 3Compute (irrational — shown to selected precision)
²√2 ≈ 1.4142135624
- 4Verify (round-trip check)
(1.41421356)^2 ≈ 2 Error: 4.44e-16
Context
Roots and powers of 2
| Expression | Value | In words |
|---|---|---|
| ²√xcurrent | 1.414214 | square root of 2 |
| ³√x | 1.259921 | cube root of 2 |
| ⁴√x | 1.189207 | 4th root of 2 |
| ⁵√x | 1.148698 | 5th root of 2 |
| x² | 4 | 2 squared |
| x³ | 8 | 2 cubed |
Field guide
Square roots, cube roots, and nth roots explained.
A root is the inverse operation of exponentiation. If 3² = 9, then √9 = 3: the square root of 9 is 3. More generally, the nth root of a number x is the value y such that y^n = x. The notation ⁿ√x represents the nth root, and this is equivalent to the fractional exponent x^(1/n).
The fractional exponent identity
ⁿ√x = x^(1/n)
²√x = x^(1/2) = x^0.5 (square root)
³√x = x^(1/3) ≈ x^0.333 (cube root)
ⁿ√x = x^(1/n) (general nth root)
This identity is why roots and exponents are inverses of each other: (x^(1/n))^n = x^(n/n) = x^1 = x. Raising a number to the power 1/n and then to the power n returns the original number.
Square roots
The square root (n = 2) of x is the positive number that, when multiplied by itself, gives x. Every positive number has two square roots, one positive and one negative (e.g., both 3 and −3 are square roots of 9). By convention, the symbol √ refers to the principal (positive) square root.
Perfect squares are integers whose square root is also an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … For all other positive integers, the square root is irrational; its decimal expansion never terminates or repeats. √2 = 1.41421356… was one of the first numbers proven irrational, by the ancient Greeks.
Cube roots
The cube root (n = 3) of x is the number that, when cubed, equals x. Cube roots behave differently from square roots in one important way: every real number (positive, negative, or zero) has exactly one real cube root. The cube root of a negative number is negative: ³√(−8) = −2 because (−2)³ = −8.
Perfect cubes are integers whose cube root is also an integer: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …
General nth roots
The nth root generalises this to any positive integer n. Key properties:
- Even n, positive x: two real roots (positive and negative principal root). By convention, ⁿ√x returns the positive root.
- Even n, negative x: no real roots; result is complex/imaginary. For example, √(−4) = 2i where i = √(−1).
- Odd n, any x: exactly one real root, with the same sign as x.
- n = 1: the first root of x is just x itself (trivial).
Imaginary and complex numbers
When you take an even root of a negative number, the result is an imaginary number. The imaginary unit i is defined as i = √(−1), so i² = −1. An even root of −x is:
ⁿ√(−x) = ⁿ√x · i (for even n, x > 0)
Example: √(−4) = √4 · i = 2i
Example: ⁴√(−16) = ⁴√16 · i = 2i
Irrational vs perfect roots
A perfect nth root occurs when ⁿ√x is an exact integer. For example:
- √144 = 12 (12² = 144): perfect square
- ³√125 = 5 (5³ = 125): perfect cube
- ⁴√256 = 4 (4⁴ = 256): perfect 4th power
When the result is not an integer, the root is irrational and its decimal expansion is infinite and non-repeating. This calculator shows up to 12 decimal places, far beyond the precision of any practical calculation, for maximum accuracy.
Common applications of roots
| Application | Root used |
|---|---|
| Length of hypotenuse (Pythagorean theorem) | √(a² + b²) |
| RMS (root mean square) value in AC circuits | √(mean of squares) |
| Standard deviation formula | √(variance) |
| Compound interest: CAGR calculation | ⁿ√(FV/PV) − 1 |
| Geometric mean of n numbers | ⁿ√(x₁×x₂×⋯×xₙ) |
| Side length from volume of a cube | ³√V |
| Radius from volume of a sphere | ³√(3V/4π) |
| Z-score in normal distribution | √variance (σ) |
How roots are computed
This calculator uses JavaScript's built-in Math.sqrt() for square roots and Math.cbrt() for cube roots, both are correctly rounded to the nearest IEEE 754 double-precision value. For higher roots,Math.pow(x, 1/n) is used, which gives results accurate to approximately 15 significant figures. The verification step computes result^n and shows the round-trip error, which is typically less than 10^(−10) for well-conditioned inputs.