Math calculator

Free root calculator

Calculate square roots, cube roots, and nth roots of any number — enter a value and root degree to see the exact result, updated live, as you type.

On this page10 sections

InputsLive

x (radicand)

The number to take the root of.

n (root degree)

Enter 2 for square root, 3 for cube root, etc.

Result

Root result

³√64 = 4

x (radicand)64

n (root degree)3

Result4

Result²16

nth root of x. Negative x with an odd integer root returns a real value.

Results are estimates. Consult a professional.

How it's calculated

How the root calculator works

The nth root of a number x is the value r such that rⁿ = x. Square roots (n=2) and cube roots (n=3) are the most common, but the formula generalises to any positive integer n. For computation, Newton's method refines an initial guess iteratively until the result converges.

ⁿ√x = x^(1/n)

√x = x^(1/2) (square root, n = 2)

∛x = x^(1/3) (cube root, n = 3)

Newton's method: rₖ₊₁ = ((n−1)·rₖ + x/rₖⁿ⁻¹) / n

Weisstein, E. W. Square Root; Nth Root. MathWorld.

Example

Worked example: √144, ∛27, and ⁴√256

Example: common roots of perfect powers

Each example below involves a perfect power, so the answer is an exact integer. For non-perfect powers the calculator returns a decimal approximation.

√144 = 144^(1/2) = 12 because 12² = 144

∛27 = 27^(1/3) = 3 because 3³ = 27

⁴√256 = 256^(1/4) = 4 because 4⁴ = 256

12, 3, 4

All three are perfect powers with exact integer roots.

Quick reference

Perfect squares and perfect cubes

Memorising common perfect squares and cubes speeds up mental maths and lets you quickly estimate roots of nearby numbers.

n	n² (perfect square)	n³ (perfect cube)
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	729
10	100	1,000

Perfect squares for n = 1–10 and perfect cubes for n = 1–10.

Practical tips

Tips for calculating roots

A few strategies make root calculations faster and more accurate.

Use exponent notation — ⁿ√x = x^(1/n) so any scientific calculator or spreadsheet can evaluate any root with the ^ or POWER function.
Factor perfect squares out — simplify √72 as √(36×2) = 6√2 rather than computing a decimal, keeping the answer exact.
Estimate by bracketing — for √50, note 7² = 49 and 8² = 64, so the answer is just above 7 (≈ 7.07).
Even roots need non-negative inputs — √x and ⁴√x are undefined for negative real x; odd roots (∛x) are defined for all real x.
Check with inverse — raise your computed root to the nth power and verify you recover the original number.

Accuracy & limits

Accuracy and limitations

Results are calculated using IEEE 754 double-precision arithmetic, giving approximately 15–16 significant decimal digits of accuracy. Even roots of negative numbers are not real and will return an error or NaN. Very large or very small inputs (beyond roughly ±10³⁰⁸) may overflow or underflow. For exact symbolic results with irrational roots, use a computer algebra system rather than a decimal calculator.

Glossary

Key terms

The value r such that rⁿ = x, written ⁿ√x or x^(1/n).

The 2nd root of x (n=2): the number r where r² = x.

The 3rd root of x (n=3): the number r where r³ = x.

An integer whose square root is also an integer (e.g. 1, 4, 9, 16, 25).

An integer whose cube root is also an integer (e.g. 1, 8, 27, 64, 125).

The symbol √ (or ⁿ√) denoting a root operation; the expression under it is called the radicand.

About

About this calculator

Part of our math calculators suite — explore all calculators.

Questions

Frequently asked questions about the free root calculator

A root calculator is a free online tool that helps you find the nth root of any number: square root (n=2), cube root (n=3), or any other root. The nth root of x is the number that, raised to the nth power, gives back x. It's just the inverse of exponentiation, written x^(1/n) under the hood. It runs entirely in your browser with instant results and no sign-up.

Because no real number squared gives a negative result — squaring always produces a non-negative number. The square root of −1 is the imaginary unit i, which lives outside the real-number system. This calculator returns only real results.

A negative number. Unlike square roots, odd-indexed roots of negatives ARE real: ∛(−27) = −3, because (−3)³ = −27. The calculator handles this correctly when n is an odd integer.

They're inverses of each other. b^n means 'multiply b by itself n times'; ⁿ√x means 'find the value that, when multiplied by itself n times, gives x'. Under the hood, ⁿ√x is just x^(1/n).