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Nth root: definition, notation, properties, and uses

An nth root of a number is a value which raised to the power n gives the number. This article explains notation, real and complex roots, properties, history, examples, and common cautions.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: May 10, 2026

Overview

An n^th root of a number r is any number k that satisfies k^n = r when multiplied by itself n times. The operation of taking an n^th root is the inverse of exponentiation: for positive integers n, if x = k^n then k is an n^th root of x. The generic expression for an n^th root is written using the radical sign with an index: the n^th root of r is written as √ $k^{n}=r$ _n r, commonly rendered in plain text as nth-root(r). For background on exponent rules that explain this inversion see related material.

Notation and the principal root

The symbol √ with an index n (often omitted when n = 2) denotes the principal n^th root. By convention the principal root of a nonnegative real number is the nonnegative real root when one exists. For example, the principal square root of 9 is 3, because 3^2 = 9. When an expression uses fractional exponents the relation x^{a/b} = (x^{a})^{1/b} = (√ $k^{n}$ _b x)^{a} links radicals and powers; this equivalence is one reason why roots are often written as x^{1/n} in algebra and calculus. See also basic treatments on radical expressions.

Real versus complex roots

Whether an n^th root exists as a real number depends on n and the radicand r. For odd n every real number r has exactly one real n^th root: for example, the cube root of -8 is -2 because (-2)^3 = -8. For even n a real n^th root exists only for r ≥ 0; negative radicands produce no real root but do have complex roots. In the complex plane any nonzero number has exactly n distinct complex n^th roots; these are evenly spaced in angle and can be described in terms of roots of unity. The collection of complex solutions explains why textbooks often distinguish the principal root from “the set of all n^th roots.” For further formal discussion see complex roots treatments.

Key properties and algebraic rules

Roots obey many familiar algebraic identities, subject to sign and domain cautions. Important relations include:

Radical–exponent equivalence: √ ${\sqrt[{n}]{r}}$ _n(x^m) = x^{m/n} = (√ ${\sqrt[{3}]{8}}=2$ _n x)^m when interpreted consistently.
Product rule (with restrictions): √ $2^{3}=8$ (ab) = √ ${\sqrt[{b}]{x^{a}}}=x^{\frac {a}{b}}=({\sqrt[{b}]{x}})^{a}=(x^{a})^{\frac {1}{b}}$ a × √ ${\sqrt {ab}}={\sqrt {a}}\times {\sqrt {b}}$ b for nonnegative a and b; if signs are mixed the identity can fail when n is even.
Quotient rule: √ ${\sqrt {\tfrac {a}{b}}}={\tfrac {\sqrt {a}}{\sqrt {b}}}$ (a/b) = (√ a)/(√ b) provided b ≠ 0 and domain conditions hold.

When manipulating symbolic expressions it is important to track principal values and the possibility of multiple roots; treating roots as exponents (x^{1/n}) helps avoid sign errors in many contexts. For related algebraic techniques see exponent rules.

History and applications

The radical sign in its modern form originated in the early modern period; the notation evolved as algebra developed in Europe. N^th roots appear across mathematics and applied fields: solving polynomial equations, constructing geometric lengths, computing mean values (the geometric mean uses roots), and in numerical methods such as root-finding algorithms. In engineering and physics fractional powers and roots describe scaling laws and material behavior. Educationally, roots are a standard part of algebra and precalculus curricula because they invert powers and enable solving equations of the form x^n = a.

Examples, distinctions and cautions

Simple numeric examples illustrate the ideas: the cube root of 8 is 2 because 2^3 = 8; the fourth root of 16 is 2 because 2^4 = 16; the cube root of -27 is -3. Distinctions to remember: "square root" usually implies the principal nonnegative root; "nth roots" in the complex sense produce n distinct solutions; and algebraic properties such as √(ab) = √a√b require attention to the signs and whether n is even. For practical computations, many calculators interpret x^{1/n} as the principal real root when available. For further reading and worked examples consult introductory resources.

Definition, Speech and Spelling

Let $n\geq 1$ be a natural number. If is a nonnegative real number, then the equation has

$x^{n}=a$

exactly one non-negative real solution. This is called the -th root of . One writes for it:

$x={\sqrt[{n\,}]{a}}$

Here one designates

${\sqrt[{n\,}]{a}}$ as root, radical or radix,
${\sqrt {\;\;}}$ as the root character,
as the root exponent,
as a radicand.

In the special case n=1 get a ${\sqrt[{1\,}]{a}}=a$ .

square and cube root

Usually the second root is called the square root or just the root and the root exponent is omitted:

${\sqrt {a}}={\sqrt[{2}]{a}}$

The root with the root exponent 3 (third root) is also called the cube root.

Example:

${\sqrt[{3}]{8}}=2$

(Say: The third root of 8 is 2 or The cube root of 8 is 2)

Mathematical basics

The following description of radixing as a right-unique root function refers to the ordered body $\mathbb{R}$ of the real numbers, that is, in a sense, to school mathematics. A more general notion of roots, encompassing the one described here, is treated in the article Adjunction (Algebra).

Connection with powers

Rooting with the root exponent and exponentiating with the exponent cancel each other. According to the above definition of the root, for all real numbers $a\geq 0$ and for all natural numbers $n\geq 1$ :

$\left({\sqrt[{n}]{a}}\right)^{n}=a$

The root exponent same effect as the exponent ${\tfrac {1}{n}}$ . Indeed, according to the rules of calculation for powers:

$\left(a^{\frac {1}{n}}\right)^{n}=a^{\frac {n}{n}}=a^{1}=a$

Therefore, radixing with the root exponent n can also be interpreted as exponentiating with the exponent 1/n:

${\sqrt[{n}]{a}}=a^{\frac {1}{n}}$

Uniqueness of roots from positive numbers

Although the problem mentioned at the beginning has two solutions with different signs for even root exponents and positive radicands, the notation with the root sign ${\sqrt[{}]{}}$ basically stands for the positive solution. For example, the equation $x^{2}=4$ has the two solutions $x=+2$ and x=-2 . However, the term ${\sqrt[{2}]{4}}$ has the value +2 and not the value -2. In general, therefore, the following applies to even-numbered root exponents

${\sqrt[{2n}]{x^{2n}}}=|x|\,.$

Roots from negative numbers

The treatment of roots from negative numbers is not uniform. It is valid for example

$(-2)^{3}=-8\,,$

and is the only real number whose third power is . In general, odd powers of negative numbers result in negative numbers again.

Regarding odd roots from negative numbers, the following positions are taken:

Roots of negative numbers are generally undefined. For example, ${\sqrt[{3}]{-8}}$ is thus undefined. The solution to the equation $x^{3}=-8$ is written as $x=-{\sqrt[{3}]{8}}$ .
Roots of negative numbers are defined when the root exponent is an odd number (3, 5, 7, ...). For odd numbers holds in general

${\sqrt[{2n+1}]{-a}}=-{\sqrt[{2n+1}]{a}}$ .

This determination is inconsistent with some properties of roots that apply to positive radicands. For example

$-2={\sqrt[{3}]{-8}}\neq {\sqrt[{6}]{(-8)^{2}}}={\sqrt[{6}]{64}}=+2.$

Also, this definition does not work with the equation ${\sqrt[{k}]{a}}=a^{\frac {1}{k}}=\exp \left({\tfrac {1}{k}}\ln(a)\right)$ , since the (natural) logarithm of negative numbers is not defined ( so must not be negative).

Roots to even exponents of negative numbers cannot be real numbers, because even powers of real numbers are never negative. There is no real number , so $x^{2}=-1$ , so you cannot $x={\sqrt[{2}]{-1}}$ find a root the real numbers. The need for roots of negative numbers led to the introduction of the complex numbers; however, there are certain difficulties with the concept of roots in the domain of the complex numbers with the unique labeling of one of the roots, see below.

Irrational roots from integers

If is a nonnegative integer and is a positive integer, then ${\sqrt[ {k}]{n}}$ either an integer or an irrational number. This is proved by applying the uniqueness of the prime factorization:

If $n\leqq 1$ , then ${\sqrt[{k}]{n}}=n$ , that is, an integer. Otherwise, there is a prime factorization $n=p_{1}^{e_{1}}\dotsm p_{r}^{e_{r}}$ with pairwise distinct primes $p_{1},\dotsc ,p_{r}$ and positive integer exponents $e_{1},\dotsc ,e_{r}$ . If all $e_{j}$ for are divisible $1\leqq j\leqq r$ by , so ${\sqrt[{k}]{n}}=p_{1}^{e_{1}/k}\dotsm p_{r}^{e_{r}/k}$ , i.e., an integer.

Now it remains to show: If there is at least one with $1\leqq j\leqq r$ such that $e_{j}$ is not divisible by , then ${\sqrt[ {k}]{n}}$ irrational. The proof of irrationality is indirect, that is, by refuting the opposite assumption as in the proof of the irrationality of the square root of 2 in Euclid, which is essentially the special case n=k=2 this proof.

Suppose ${\sqrt[ {k}]{n}}$ were rational. Then you could write the number as a fraction of two natural numbers and

${\sqrt[{k}]{n}}={\frac {a}{b}}$ .

By exponentiating the equation you get

$n={\frac {a^{k}}{b^{k}}}$

and from this follows

$nb^{k}=a^{k}$ .

The prime factor $p_{j}$ occurs in $a^{k}$ or $b^{k}$ times as often as in resp. in any case in a multiplicity divisible by , although of course the occurrence 0 times is also allowed. In it occurs presuppositionally in the multiplicity $e_{j}$ not divisible by . So on the left-hand side of this equation it does not occur in a manifold divisible by , but on the right-hand side it does, and we get a contradiction to the uniqueness of the prime factorization. Therefore, is ${\sqrt[ {k}]{n}}$ irrational.

The root laws

The calculation rules for roots follow from those for powers.

For positive numbers and and the following computational laws hold $n,m,k\in \mathbb {N}$ :

Product rule: ${\sqrt[{n}]{a}}\cdot {\sqrt[{n}]{b}}={\sqrt[{n}]{a\cdot b}}$
Quotient rule: ${\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}={\sqrt[{n}]{\frac {a}{b}}}$
"Nesting rule" or iteration rule: ${\sqrt[ {m}]{{\sqrt[ {n}]{a}}}}={\sqrt[ {m\cdot n}]{a}}$
Definition for fractional exponents: $a^{\frac {k}{n}}={\sqrt[{n}]{a^{k}}}=\left({\sqrt[{n}]{a}}\right)^{k}$
Definition for negative exponents: $a^{-{\frac {k}{n}}}={\frac {1}{a^{\frac {k}{n}}}}$
With the same radicand, ${\sqrt[{m}]{a}}\cdot {\sqrt[{n}]{a}}=a^{{\frac {1}{m}}+{\frac {1}{n}}}={\sqrt[{mn}]{a^{m+n}}}$

For negative numbers and these calculation laws may only be applied if and are odd numbers. For complex numbers, they are to be avoided altogether, or equality applies only if the adjoint values are chosen appropriately. In other words, if any roots (e.g., only principal values) are chosen on the left side in an example, there are suitable minor values for the right side that satisfy equality - left and right sides differ by one unit root.

Limit values

The following limits apply:

$\lim _{n\rightarrow \infty }{\sqrt[{n}]{a}}=1$ for
$\lim _{n\rightarrow \infty }{\sqrt[{n}]{n}}=1$

This follows from the inequality $n<\left(1+{\sqrt[{2}]{\tfrac {2}{n}}}\right)^{n}$ , which can be shown using the binomial theorem.

$\lim _{n\to \infty }{\sqrt[{n}]{n^{k}}}=1$ , where an arbitrary but fixed natural number.
$\lim _{n\rightarrow \infty }{\frac {\ln(n)}{n}}=0$ ,

as can be seen from the exponential representation of . ${\sqrt[{n}]{n}}$

Root functions

Shape functions

$f\colon \mathbb {R} _{0}^{+}\to \mathbb {R} _{0}^{+},x\mapsto {\sqrt[{n}]{x}}$ or more generally $x\mapsto {\sqrt[{n}]{x^{m}}}$

are called root functions. They are power functions, it holds ${\sqrt[{n}]{x^{m}}}=x^{\frac {m}{n}}$ .

Questions and Answers

Q: What is an n-th root?

A: An n-th root of a number r is a number which, if multiplied by itself n times, produces the number r.

Q: How is an n-th root written?

A: An n-th root of a number r is written as r^(1/n).

Q: What are some examples of roots?

A: If the index (n) is 2, then the radical expression is a square root. If it is 3, it is a cube root. Other values of n are referred to using ordinal numbers such as fourth root and tenth root.

Q: What does the product property of a radical expression state?

A: The product property of a radical expression states that sqrt(ab) = sqrt(a) x sqrt(b).

Q: What does the quotient property of a radical expression state?

A: The quotient property of a radical expression states that sqrt(a/b) = (sqrt(a))/(sqrt(b)), where b != 0.

Q: What other terms can be used to refer to an n-th root?

A: An n-th root can also be referred to as a radical or radical expression.

Author

AlegsaOnline.com - n th root - Leandro Alegsa

Creation date: November 13, 2022
Last updated: May 10, 2026

URL: https://en.alegsaonline.com/art/71324