# Nth root

An **nth root** of a number a is a number b whose nth power equals a, that is, a number satisfying \( b^{n} = a \) for a positive integer n. For example, −3 is a 5th root of −243 because \( (-3)^{5} = -243 \).<sup>[1](https://openstax.org/books/algebra-and-trigonometry/pages/1-3-radicals-and-rational-exponents)</sup> Over the real numbers the situation depends on the parity of n; over the complex numbers every nonzero number has exactly n distinct nth roots, arranged as the vertices of a regular polygon.<sup>[2](https://www.ms.uky.edu/~droyster/courses/fall06/PDFs/Chapter04.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | If \( b^{n} = a \), then b is an nth root of a; the principal nth root is written \( \sqrt[n]{a} \), with n called the index of the radical.<sup>[3](https://openstax.org/books/elementary-algebra/pages/9-7-higher-roots)</sup> |
| Real roots | When n is even, \( \sqrt[n]{a} \) is real only for \( a \ge 0 \); when n is odd, \( \sqrt[n]{a} \) is real for all real a.<sup>[3](https://openstax.org/books/elementary-algebra/pages/9-7-higher-roots)</sup> |
| Complex roots | Every nonzero complex number has exactly n distinct complex nth roots \( z_{k} = \sqrt[n]{\lvert z \rvert}\left(\cos\frac{\mathrm{Arg}(z) + 2\pi k}{n} + i\sin\frac{\mathrm{Arg}(z) + 2\pi k}{n}\right) \), k = 0, …, n − 1.<sup>[2](https://www.ms.uky.edu/~droyster/courses/fall06/PDFs/Chapter04.pdf)</sup> |
| Geometry | The n complex roots are the vertices of a regular n-gon inscribed in the circle of radius \( \sqrt[n]{\lvert z \rvert} \) centered at the origin.<sup>[2](https://www.ms.uky.edu/~droyster/courses/fall06/PDFs/Chapter04.pdf)</sup> |
| Roots of unity | The nth roots of 1 are \( 1, \omega_{n}, \omega_{n}^{2}, \ldots, \omega_{n}^{n-1} \) with \( \omega_{n} = \cos\frac{2\pi}{n} + i\sin\frac{2\pi}{n} \).<sup>[4](https://www.math.cuhk.edu.hk/course_builder/2122/math1050a/1050a-l05h-2.pdf)</sup> |
| Numerical extraction | Newton's method applied to \( x^{p} - N \) gives the iteration \( x_{n+1} = \frac{1}{p}\left[\frac{(p-1)x_{n}^{p} + N}{x_{n}^{p-1}}\right] \).<sup>[5](https://www.annamalaiuniversity.ac.in/studport/download/engg/math/resources/Dr%20ST-BS401-Numerical%20Methods-Module-4%265.pdf)</sup> |

## Real nth roots and the radical

The radical notation \( \sqrt[n]{a} \) names the principal nth root. For a real number a with at least one nth root, the principal root is the root with the same sign as a: for \( a \ge 0 \) and even n it is the nonnegative root, and for odd n it is the unique real root, which shares the sign of a.<sup>[1](https://openstax.org/books/algebra-and-trigonometry/pages/1-3-radicals-and-rational-exponents)</sup> The index n is written in the hook of the radical sign; it is omitted for the square root, and \( \sqrt[3]{a} \) is read as the cube root.<sup>[3](https://openstax.org/books/elementary-algebra/pages/9-7-higher-roots)</sup>

Parity controls existence over the reals. Even roots of negative numbers are not real numbers, while odd roots of negative numbers are real: \( \sqrt[5]{-32} = -2 \) because \( (-2)^{5} = -32 \).<sup>[3](https://openstax.org/books/elementary-algebra/pages/9-7-higher-roots)</sup><sup> • </sup><sup>[1](https://openstax.org/books/algebra-and-trigonometry/pages/1-3-radicals-and-rational-exponents)</sup> This is why equations such as \( x^{2} = -1 \) have no real solution but led to the extension of the number system to the complex numbers, where roots always exist for nonzero values.

## Complex nth roots and De Moivre's theorem

The tool for finding all roots of a complex number is De Moivre's theorem, which states that \( \bigl(r(\cos\theta + i\sin\theta)\bigr)^{n} = r^{n}(\cos n\theta + i\sin n\theta) \).<sup>[6](https://a-warded.org/open-book/university-complex-analysis/lesson-5)</sup> Because sine and cosine are periodic with period \( 2\pi \), the argument of a complex number is determined only up to multiples of \( 2\pi \), and each choice produces a different root. For a nonzero complex number z, the n roots are

\[ z_{k} = \sqrt[n]{\lvert z \rvert}\left(\cos\frac{\mathrm{Arg}(z) + 2\pi k}{n} + i\sin\frac{\mathrm{Arg}(z) + 2\pi k}{n}\right), \qquad k = 0, 1, \ldots, n-1. \]

Only these n values of k give distinct results; larger k repeats earlier roots.<sup>[2](https://www.ms.uky.edu/~droyster/courses/fall06/PDFs/Chapter04.pdf)</sup> Equivalently, once one root ζ is found, the full set is \( \zeta, \zeta\omega_{n}, \zeta\omega_{n}^{2}, \ldots, \zeta\omega_{n}^{n-1} \), obtained by multiplying by the roots of unity.<sup>[4](https://www.math.cuhk.edu.hk/course_builder/2122/math1050a/1050a-l05h-2.pdf)</sup>

A worked example shows the structure. The fifth roots of \( a = -1 - i \) are \( \sqrt[5]{\sqrt{2}} \) times the numbers at angles \( (-3\pi/4 + 2\pi k)/5 \) for k = 0 through 4, giving five points equally spaced by \( 2\pi/5 \) radians; plotted in the complex plane they form the vertices of a regular pentagon.<sup>[2](https://www.ms.uky.edu/~droyster/courses/fall06/PDFs/Chapter04.pdf)</sup>

## Roots of unity

Setting z = 1 in the root formula gives the **nth roots of unity**, the n solutions of \( z^{n} = 1 \). They are

\[ \omega_{n}^{k} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}, \qquad k = 0, 1, \ldots, n-1, \]

and they are the vertices of a regular n-gon inscribed in the unit circle with 1 at one vertex.<sup>[2](https://www.ms.uky.edu/~droyster/courses/fall06/PDFs/Chapter04.pdf)</sup> The generator \( \omega_{n} = \cos\frac{2\pi}{n} + i\sin\frac{2\pi}{n} \) satisfies \( \omega_{n}^{n} = \cos 2\pi + i\sin 2\pi = 1 \), so its powers run through the whole set.<sup>[4](https://www.math.cuhk.edu.hk/course_builder/2122/math1050a/1050a-l05h-2.pdf)</sup>

## The logarithm viewpoint and multivaluedness

In complex analysis, roots are a special case of complex powers. Once a branch L of the logarithm is fixed on a domain, powers are defined by \( z^{\alpha} = \exp(\alpha L(z)) \), and the nth roots of \( z \neq 0 \) are

\[ z^{1/n} = \exp\left(\frac{1}{n}\log z\right) = \lvert z \rvert^{1/n}\exp\left(i\,\frac{\mathrm{Arg}(z) + 2\pi k}{n}\right), \qquad k = 0, \ldots, n-1. \]

The multivaluedness comes from the logarithm: because \( \exp \) is many-to-one, its inverse takes infinitely many values differing by multiples of \( 2\pi i \).<sup>[7](https://fac.iitg.ac.in/rksri/MA547%20Complex%20Analysis%20lecture%20notes%202025.pdf)</sup><sup> • </sup><sup>[8](https://complexanalysis.org/web/sec_complex-log.html)</sup> To obtain a single-valued function one chooses a principal value, for example \( \mathrm{Log}(z) = \ln\lvert z\rvert + i\,\mathrm{Arg}(z) \) with \( -\pi < \mathrm{Arg}(z) \le \pi \), and a branch cut along which the function is discontinuous, conventionally the negative real axis.<sup>[8](https://complexanalysis.org/web/sec_complex-log.html)</sup><sup> • </sup><sup>[9](https://dlmf.nist.gov/4.2)</sup> This is the same choice that makes \( \sqrt{z} \) a single-valued function on the cut plane, and the multivalued logarithm becomes single-valued on a [Riemann surface](https://www.edgepedia.org/riemann-surface) built by gluing copies of the slit plane.<sup>[7](https://fac.iitg.ac.in/rksri/MA547%20Complex%20Analysis%20lecture%20notes%202025.pdf)</sup>

## Numerical extraction

Exact roots rarely exist as decimals, so computation relies on iterative methods. [Newton's method](https://www.edgepedia.org/newtons-method) applied to \( f(x) = x^{p} - N \) produces the update

\[ x_{n+1} = \frac{1}{p}\left[\frac{(p-1)x_{n}^{p} + N}{x_{n}^{p-1}}\right], \]

which converges to the pth root of N.<sup>[5](https://www.annamalaiuniversity.ac.in/studport/download/engg/math/resources/Dr%20ST-BS401-Numerical%20Methods-Module-4%265.pdf)</sup> For the square root case this reduces to the familiar averaging iteration \( x_{n+1} = \frac{1}{2}\left(x_{n} + \frac{N}{x_{n}}\right) \). Newton's method converges quadratically near a simple root, meaning the number of correct digits roughly doubles per step, which makes it far more efficient than bisection or the secant method when a good starting value is available.<sup>[10](https://www.d.umn.edu/~mhampton/NA/Roots/)</sup> Floating-point hardware and libraries implement square root as a correctly rounded basic operation under the [IEEE 754](https://www.edgepedia.org/ieee-754) standard, which specifies formats, the square root operation, and exception handling for computer arithmetic.<sup>[11](https://standards.ieee.org/ieee/315/6210/)</sup>

## Where nth roots appear

The nth root operation defines the **geometric mean** of n positive numbers \( X_{1}, \ldots, X_{n} \), namely \( \left(\prod_{i=1}^{n} X_{i}\right)^{1/n} \), a standard summary statistic for quantities that multiply together, such as growth rates.<sup>[12](https://itl.nist.gov/div898/software/dataplot/refman2/auxillar/geommean.htm)</sup> Rational exponents express roots in exponent form, \( x^{m/n} = (x^{m})^{1/n} \) for \( x > 0 \), connecting radicals to the general power function \( x^{\alpha} = e^{\alpha \ln x} \).<sup>[13](https://math.ou.edu/~npetrov/math4103_s16.html)</sup>

## History

Approximating roots predates the notation by millennia. The Old Babylonian tablet [YBC 7289](https://www.edgepedia.org/ybc-7289), dated to roughly 1800–1600 BCE, shows a square of side 30 with its diagonal labeled 42;25,35 in base sixty, exactly 30 times the scribe's estimate \( \sqrt{2} \approx 1;24,51,10 \), which equals about 1.414213 and is off by less than one part in two million.<sup>[14](https://artwaste.land/strata/the-diagonal-they-already-knew/)</sup> The radical symbol itself is younger: the oldest notation for radicals dates to about 1480, when a dot placed before the radicand signified a square root, two dots a fourth root, and three dots a cube root; the modern convention of writing the exponent above the opening of the radical sign became customary only at the beginning of the 18th century.<sup>[15](https://academic.oup.com/princeton-scholarship-online/book/15483/chapter-abstract/170145042)</sup><sup> • </sup><sup>[16](https://encyclopediaofmath.org/wiki/Mathematical_symbols)</sup>

## References

1. [Radicals and Rational Exponents, OpenStax Algebra and Trigonometry](https://openstax.org/books/algebra-and-trigonometry/pages/1-3-radicals-and-rational-exponents)
2. [Chapter 4: Roots of Complex Numbers, University of Kentucky course notes (MATH 6101)](https://www.ms.uky.edu/~droyster/courses/fall06/PDFs/Chapter04.pdf)
3. [Higher Roots, OpenStax Elementary Algebra, section 9.7](https://openstax.org/books/elementary-algebra/pages/9-7-higher-roots)
4. [De Moivre's Theorem and Roots of Unity, MATH1050, The Chinese University of Hong Kong](https://www.math.cuhk.edu.hk/course_builder/2122/math1050a/1050a-l05h-2.pdf)
5. [Newton Scheme of Iteration for Finding the pth Root, Numerical Methods, Annamalai University](https://www.annamalaiuniversity.ac.in/studport/download/engg/math/resources/Dr%20ST-BS401-Numerical%20Methods-Module-4%265.pdf)
6. [Products, Powers, and Roots of Complex Numbers, A-Warded Complex Analysis](https://a-warded.org/open-book/university-complex-analysis/lesson-5)
7. [MA547 Complex Analysis lecture notes, IIT Guwahati](https://fac.iitg.ac.in/rksri/MA547%20Complex%20Analysis%20lecture%20notes%202025.pdf)
8. [The Complex Logarithm, complexanalysis.org](https://complexanalysis.org/web/sec_complex-log.html)
9. [DLMF §4.2 Definitions: Logarithm, Exponential, Powers, NIST](https://dlmf.nist.gov/4.2)
10. [Roots of Nonlinear Equations, Introduction to Numerical Methods, University of Minnesota Duluth](https://www.d.umn.edu/~mhampton/NA/Roots/)
11. [IEEE 754-2019, IEEE Standard for Floating-Point Arithmetic](https://standards.ieee.org/ieee/315/6210/)
12. [Geometric Mean, NIST Dataplot Reference Manual](https://itl.nist.gov/div898/software/dataplot/refman2/auxillar/geommean.htm)
13. [Math 4103 lecture schedule, University of Oklahoma](https://math.ou.edu/~npetrov/math4103_s16.html)
14. [YBC 7289: the square root of 2 in Babylonian base sixty](https://artwaste.land/strata/the-diagonal-they-already-knew/)
15. [The Timid Symbol, Enlightening Symbols, Oxford Academic](https://academic.oup.com/princeton-scholarship-online/book/15483/chapter-abstract/170145042)
16. [Mathematical symbols, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Mathematical_symbols)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Elementary and formal arithmetic › Elementary arithmetic operations*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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