Nth Root Calculator – Free Calculator

Understanding the nth root calculator — concept and principle

The nth root of a number x is the value that, multiplied by itself n times, gives back x. The square root (n = 2) and cube root (n = 3) are the familiar cases, but the same idea works for any n. Roots are used to undo powers: finding the side of a square or cube from its area or volume, the average yearly growth rate from a total growth factor, or the geometric mean of several values.

A root runs a power backwards: 3 used four times makes 81, so ⁴√81 = 3.

How it is calculated

Formula: ⁿ√x = x^(1/n).
Negative x with odd n: the calculator computes −((−x)^(1/n)), so the cube root of −27 is −3.
Negative x with even n: there is no real root, and the result is shown as not a number.

For positive x the calculator returns the principal (positive) root. Both inputs are pure numbers; n need not be a whole number, although roots are normally taken with whole n of 2 or more.

The principle

Taking a root is the inverse of raising to a power. If r = x^(1/n), then rⁿ = (x^(1/n))ⁿ = x^(n/n) = x, using the power-of-a-power rule. An even power can never be negative, because a negative number multiplied by itself an even number of times is positive; that is why even roots of negative numbers have no real value. An odd power keeps the sign, so every real number has exactly one real odd root. For a positive x and even n there are two real roots, +r and −r, and by convention the symbol ⁿ√ means the positive one.

Worked example

Inputs: number x = 81, root n = 4.
Compute 81^(1/4) = 81^0.25.
Check by factoring: 81 = 3 × 3 × 3 × 3 = 3⁴.
Therefore ⁴√81 = 3, and indeed 3⁴ = 81.

Each point is the calculator re-run with only “Root (n)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Root (n)	ⁿ√x
1	81
2	9
3	4.326748711
4	3
5	2.408224685
6	2.080083823
8	1.732050808

How to read the result

For x greater than 1, a larger n gives a smaller root that approaches 1; for x between 0 and 1, the root grows toward 1. A whole-number result means x is a perfect nth power. In finance, the nth root of a total growth factor gives the average growth per period: if money doubles in 10 years, the 10th root of 2, about 1.0718, means roughly 7.18% per year.

Common mistakes

Dividing by n instead of taking a root (⁴√81 is 3, not 20.25).
Forgetting the negative root of x² = 81: the equation has two solutions, 9 and −9, while √81 itself is 9.
Entering a negative number with an even n and expecting a real answer.
Rounding an irrational root too early and then raising it back to the power, which magnifies the error.

Keep in mind: Most roots, such as √2 or ³√10, are irrational, so the displayed value is a rounded approximation from floating-point arithmetic. Keep the full value in later steps and round only the final answer.

Frequently asked questions

How do I calculate a cube root without a calculator?

Find perfect cubes on either side (for example 27 and 64 around 40), then refine by trial: 3.4³ ≈ 39.3 and 3.42³ ≈ 40.0, so ³√40 ≈ 3.42.

Is the square root of a number always smaller than the number?

Only when the number is greater than 1. For 0.25 the square root is 0.5, which is larger.

What is the relation between roots and fractional exponents?

They are the same operation written differently: ⁿ√x = x^(1/n), and ⁿ√(xᵐ) = x^(m/n).

Sources: NIST Digital Library of Mathematical Functions, Chapter 4: Elementary Functions (§4.2, powers and roots) · IEEE 754-2019, Standard for Floating-Point Arithmetic

Checked 8 Nov 2026. Standard formulas, computed in your browser with double-precision numbers.

Your numbers