Square root, cube root or any nth root of any number — with the decimal value, simplified radical form, equivalent fractional-exponent form, and step-by-step working.
Pick a root type, enter the radicand, read the result.
√72 (decimal)
Square root
8.4853
x^(1/2)
Cube root
4.1602
x^(1/3)
nth root
2.9130
x^(1/4)
Inverse (x^n)
72
(ⁿ√x)^n verification
| Form | Notation | Example |
|---|---|---|
| Square root | √x = x^(1/2) | √16 = 4 |
| Cube root | ∛x = x^(1/3) | ∛27 = 3 |
| Fourth root | ⁴√x = x^(1/4) | ⁴√81 = 3 |
| Reciprocal | 1 / ⁿ√x | 1/√2 ≈ 0.707 |
| Simplification | √(a²·b) = a√b | √50 = 5√2 |
| Product | √a · √b = √(a·b) | √3 · √12 = √36 = 6 |
The Definition
The nth root of x, written ⁿ√x, is the number y satisfying yⁿ = x. Equivalently it's the fractional exponent x^(1/n) — every root is a special case of an exponent. The 2nd root (square root) and 3rd root (cube root) get their own symbols (√ and ∛); higher roots use the radical sign with index: ⁴√x, ⁵√x, and so on. To simplify a root, find perfect-nth-power factors and pull them out: √(a²·b) = a√b. The number left under the radical (the radicand) should contain no nth-power factors.
Working for your inputs
About This Tool
A root calculator computes the nth root of a number — the value y such that yⁿ = x. The two most common are the square root (√x) and the cube root (∛x), but you can ask for any positive integer index: ⁴√x, ⁵√x, ¹⁰√x, etc. The tool returns the decimal value, the simplified radical form (for square roots), and shows the equivalent fractional exponent form x^(1/n).
Roots are everywhere in mathematics, science and engineering. The Pythagorean theorem gives c = √(a² + b²). The quadratic formula uses a square root. Geometric means, standard deviations, compound-interest rates and even the period of a pendulum (T = 2π√(L/g)) all involve roots. In physics, the RMS (root mean square) of a signal is literally the square root of the mean of the squares.
Mathematically, this calculator uses Math.sqrt, Math.cbrt and Math.pow(x, 1/n) internally, plus an integer-factorisation routine to simplify square roots into their canonical a√b form when b is small. Results are exact up to IEEE 754 double precision — about 15–17 significant digits. The calculator handles negative radicands for odd roots (∛(−8) = −2) and flags them as undefined for even roots in the reals.
Use this free root calculator for homework, physics, engineering, financial growth calculations, or whenever you need the inverse of a power. All computation runs locally — no sign-up, no tracking.
Any Index n
Square, cube, or any positive-integer index — your choice.
Simplified Radical Form
Square roots factored — e.g. √72 → 6√2, not just 8.4853.
Exponent Equivalent
Every result is shown as the equivalent fractional exponent x^(1/n).
Inverse Check
Confirms (ⁿ√x)^n = x — a built-in sanity verification.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Curve Visualisation
Plot of y = ⁿ√x with your point marked — see how roots grow.
Pick a root type, type the radicand, read the answer.
Choose square root, cube root, or custom nth root. The two presets cover almost every everyday case.
Type the number under the root, x. Negative values are accepted for odd roots (cube, fifth, etc.) but undefined in the reals for even roots.
If you picked "custom", type the index n. Any integer ≥ 2 works. Common: n = 4 for fourth root, n = 5 for fifth root.
The summary card shows the decimal value. For square roots, the meta line also shows the simplified radical form (e.g. 6√2).
Stat tiles show √x, ∛x, and ⁿ√x side-by-side — useful for spotting how the answer changes with n.
The formula card shows the simplification and verifies (ⁿ√x)^n = x — exactly the steps you'd write in a homework answer.
Everything you need to know about roots, radicands and simplified radical form.
The nth root of a number x, written ⁿ√x, is the value y such that yⁿ = x. So the 2nd root (square root) of 25 is 5 because 5² = 25; the 3rd root (cube root) of 27 is 3 because 3³ = 27. Roots are the inverse operation of exponentiation.
Odd roots of negative numbers are valid real numbers: ∛(−8) = −2 because (−2)³ = −8. Even roots of negative numbers are not real — there's no real number y with y² = −1, for example. They exist in the complex numbers (√(−1) = i), but most calculators (this one included) flag them as undefined.
Every root is a fractional exponent: ⁿ√x = x^(1/n). So √x = x^0.5, ∛x = x^(1/3), and ⁴√x = x^0.25. This identity is why your calculator can compute any root via the power function. The exponent laws all carry over: ⁿ√(xy) = ⁿ√x · ⁿ√y, and (ⁿ√x)^m = x^(m/n) = ⁿ√(x^m).
Find the largest perfect-square factor of the radicand and pull it out: √72 = √(36·2) = √36 · √2 = 6√2. The number left under the radical (here 2) is the simplified radicand. This calculator does this automatically for square roots up to reasonable size.
Only when the radicand is a perfect square. √4 = 2 is rational; √9 = 3 is rational; but √2, √3, √5, √6, √7, √8, √10 are all irrational — they cannot be written as a fraction p/q and have non-repeating, non-terminating decimal expansions. The classic proof that √2 is irrational (by contradiction) is one of the oldest results in mathematics.
Every positive number has two square roots: a positive one and a negative one. For example, both 3 and −3 satisfy x² = 9. The principal square root, written √x, is by convention the positive (or zero) one. When solving x² = 9 you write x = ±√9 = ±3 — both roots, but the symbol √ itself denotes only the positive one.
Because 0 is its own negative: +0 = −0 = 0. So while positive numbers have two distinct square roots (+r and −r), zero has only one square root, namely 0 itself. The same is true of higher even roots of zero.
The calculator uses JavaScript's Math.sqrt, Math.cbrt and Math.pow, implemented in IEEE 754 double-precision floating-point. Accuracy is about 15–17 significant digits — far more than any real-world measurement needs. For arbitrary-precision results (hundreds of digits) you would need a big-decimal library.
The inverse check computes (ⁿ√x)^n and confirms it equals x (within floating-point error). It's a sanity check that the root was computed correctly — and a reminder that roots and powers are inverse operations.
Use the identity ⁿ√(a/b) = ⁿ√a / ⁿ√b. So √(9/16) = √9 / √16 = 3/4. To rationalise (move a root out of the denominator), multiply numerator and denominator by the same root: 1/√2 = √2 / 2. The calculator works in decimals — convert your fraction first.
None mathematically — they mean the same thing. √x and x^(1/2) are notational variants of the same operation. The radical symbol √ is older and visually clearer for handwritten work; the fractional-exponent form x^(1/n) generalises more naturally (you can write x^(2/3) but "two-thirds-root of x" is awkward).
Roots appear constantly. Pythagoras: c = √(a² + b²). The quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. Standard deviation: σ = √variance. Pendulum period: T = 2π√(L/g). Compound interest doubling: r = ⁿ√(P₁/P₀) − 1. RMS voltage in AC electronics. Geometric mean of n positive numbers is their nth root.