# nth root

Calculation results
Subtraction (−)
minuendsubtrahend = difference
Multiplication (×)
multiplicand × multiplier = product
Division (÷)
dividend ÷ divisor = quotient
Exponentiation
baseexponent = power
nth root (√)
Logarithm
logbase(power) = exponent
Roots of numbers 0 through 10. Line labels = x. X axis = n. Y axis = nth root of x.

In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals x

$r^n = x,$

where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred to using ordinal numbers, as in fourth root, twentieth root, etc.

For example:

• 2 is a square root of 4, since 22 = 4.
• −2 is also a square root of 4, since (−2)2 = 4.

A real number or complex number has n roots of degree n. While the roots of 0 are not distinct (all equaling 0), the n nth roots of any other real or complex number are all distinct. If n is even and the radicand is real and positive, one of its nth roots is positive, one is negative, and the rest are complex but not real; if n is even and the radicand is real and negative, none of the nth roots is real. If n is odd and the radicand is real, one nth root is real and has the same sign as the radicand , while the other roots are not real.

Roots are usually written using the radical symbol or radix $\sqrt{\,\,}$ or $\surd{}$, with $\sqrt{x}\!\,$ or $\surd x$ denoting the square root, $\sqrt[3]{x}\!\,$ denoting the cube root, $\sqrt[4]{x}$ denoting the fourth root, and so on. In the expression $\sqrt[n]{x}$, n is called the index, $\sqrt{\,\,}$ is the radical sign or radix, and x is called the radicand. When a number is presented under the radical symbol, it must return only one result like a function, so a non-negative real root, called the principal nth root, is preferred rather than others. An unresolved root, especially one using the radical symbol, is often referred to as a surd[1] or a radical.[2] Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression.

In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction:

$\sqrt[n]{x} \,=\, x^{1/n}$

Roots are particularly important in the theory of infinite series; the root test determines the radius of convergence of a power series. Nth roots can also be defined for complex numbers, and the complex roots of 1 (the roots of unity) play an important role in higher mathematics. Galois theory can be used to determine which algebraic numbers can be expressed using roots, and to prove the Abel-Ruffini theorem, which states that a general polynomial equation of degree five or higher cannot be solved using roots alone; this result is also known as "the insolubility of the quintic".

## History

The origin of the root symbol √ is largely speculative. Some sources imply that the symbol was first used by Arabic mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī (1421–1486). Legend has it that it was taken from the Arabic letter "ج‎" (ǧīm, pronunciation: /im/), which is the first letter in the Arabic word "جذر‎" (jadhir, meaning "root"; pronunciation: /ˈɑːðir/).[3] However, many scholars, including Leonhard Euler,[4] believe it originates from the letter "r", the first letter of the Latin word "radix" (meaning "root"), referring to the same mathematical operation. The symbol was first seen in print without the vinculum (the horizontal "bar" over the numbers inside the radical symbol) in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician.

The term surd traces back to al-Khwārizmī (c. 825), who referred to rational and irrational numbers as audible and inaudible, respectively. This later led to the Arabic word "أصم‎" (asamm, meaning "deaf" or "dumb") for irrational number being translated into Latin as "surdus" (meaning "deaf" or "mute"). Gerard of Cremona (c. 1150), Fibonacci (1202), and then Robert Recorde (1551) all used the term to refer to unresolved irrational roots.[5]

## Definition and notation

The four 4th roots of −1,
none of which is real
The three 3rd roots of −1,
one of which is a negative real

The nth root of a number x, where n is a positive integer, is a number r whose nth power is x:

$r^n = x.\!\,$

Every positive real number x has a single positive nth root, which is written $\sqrt[n]{x}$. For n equal to 2 this is called the square root and the n is omitted. The nth root can also be represented using exponentiation as x1/n.

For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root. For odd values of n, every negative number x has a real negative nth root. For example, −2 has a real 5th root, $\sqrt[5]{-2} \,= -1.148698354\ldots$ but −2 does not have any real 6th roots.

Every non-zero number x, real or complex, has n different complex number nth roots including any positive or negative roots, see complex roots below. The nth root of 0 is 0.

For most numbers[quantify], an nth root is irrational. For example,

$\sqrt{2} = 1.414213562\ldots$

All nth roots of integers, or in fact of any algebraic number, are algebraic.

For the extension of powers and roots to indices that are not positive integers, see exponentiation.

The character codes for the radical symbols are

Read Character Unicode ASCII URL HTML (others)
Square root U+221A &#8730; %E2%88%9A &radic;
Cube root U+221B &#8731; %E2%88%9B
Fourth root U+221C &#8732; %E2%88%9C

### Square roots

The graph y = $\pm \sqrt{x}$.

The square root of a number x is that number r which, when squared, becomes x:

$r^2 = x.\!\,$

Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign:

$\sqrt{25} = 5.\!\,$

Since the square of every real number is a positive real number, negative numbers do not have real square roots. However, every negative number has two imaginary square roots. For example, the square roots of −25 are 5i and −5i, where i represents a square root of −1.

### Cube roots

The graph y = $\sqrt[3]{x}$.

A cube root of a number x is a number r whose cube is x:

$r^3 = x.\!\,$

Every real number x has exactly one real cube root, written $\sqrt[3]{x}$. For example,

$\sqrt[3]{8}\,=\,2\quad\text{and}\quad\sqrt[3]{-8}\,= -2.$

Every real number has two additional complex cube roots (see complex roots below).

## Identities and properties

Every positive real number has a positive nth root and the rules for operations with such surds are straightforward:

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

Using the exponent form as in $x^{1/n}$ normally makes it easier to cancel out powers and roots.

$\sqrt[n]{a^m} = \left(a^m\right)^{\frac{1}{n}} = a^{\frac{m}{n}}.$

Problems can occur when taking the nth roots of negative or complex numbers. For instance:

$\sqrt{-1}\times\sqrt{-1} = -1$

whereas

$\sqrt{-1 \times -1} = 1$

when taking the principal value of the roots. See failure of power and logarithm identities in the exponentiation article for more details.

## Simplified form of a radical expression

A radical expression is said to be in simplified form if[6]

1. There is no factor of the radicand that can be written as a power greater than or equal to the index.
2. There are no fractions under the radical sign.
3. There are no radicals in the denominator.

For example, to write the radical expression $\sqrt{\tfrac{32}{5}}$ in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:

$\sqrt{\tfrac{32}{5}} = \sqrt{\tfrac{16 \cdot 2}{5}} = 4 \sqrt{\tfrac{2}{5}}$

Next, there is a fraction under the radical sign, which we change as follows:

$4 \sqrt{\tfrac{2}{5}} = \frac{4 \sqrt{2}}{\sqrt{5}}$

Finally, we remove the radical from the denominator as follows:

$\frac{4 \sqrt{2}}{\sqrt{5}} = \frac{4 \sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{4 \sqrt{10}}{5}$

When there is a denominator involving surds it may be possible to find a factor to multiply both numerator and denominator by to simplify the expression. For instance using the factorization of the sum of two cubes:

$\frac{1}{\sqrt[3]{a}+\sqrt[3]{b}} = \frac{\sqrt[3]{a^2}-\sqrt[3]{ab}+\sqrt[3]{b^2}}{(\sqrt[3]{a}+\sqrt[3]{b})(\sqrt[3]{a^2}-\sqrt[3]{ab}+\sqrt[3]{b^2})} = \frac{\sqrt[3]{a^2}-\sqrt[3]{ab}+\sqrt[3]{b^2}}{a+b} \,.$

Simplifying radical expressions involving nested radicals can be quite difficult. It is not immediately obvious for instance that:

$\sqrt{3+2\sqrt{2}} = 1+\sqrt{2}\,$

## Infinite series

The radical or root may be represented by the infinite series:

$(1+x)^{s/t} = \sum_{n=0}^\infty \frac{\prod_{k=0}^{n-1} (s-kt)}{n!t^n}x^n$

with $|x|<1$. This expression can be derived from the binomial series.

## Computing principal roots

The nth root of an integer is not always an integer, and if it is not an integer then it is not a rational number. For instance, the fifth root of 34 is

$\sqrt[5]{34} = 2.024397458 \ldots,$

where the dots signify that the decimal expression does not end after any finite number of digits. Since in this example the digits after the decimal never enter a repeating pattern, the number is irrational.

The nth root of a number A can be computed by the nth root algorithm, a special case of Newton's method. Start with an initial guess x0 and then iterate using the recurrence relation

$x_{k+1} = \frac{1}{n} \left({(n-1)x_k +\frac{A}{x_k^{n-1}}}\right)$

until the desired precision is reached.

Depending on the application, it may be enough to use only the first Newton approximant:

$\sqrt[n]{x^n+y} \approx x + \frac{y}{n x^{n-1}}.$

For example, to find the fifth root of 34, note that 25 = 32 and thus take x = 32 and y = 2 in the above formula. This yields

$\sqrt[5]{34} = \sqrt[5]{32 + 2} \approx 2 + \frac{2}{5 \cdot 16} = 2.025.$

The error in the approximation is only about 0.03%.

Newton's method can be modified to produce a generalized continued fraction for the nth root which can be modified in various ways as described in that article. For example:

$\sqrt[n]{z} = \sqrt[n]{x^n+y} = x+\cfrac{y} {nx^{n-1}+\cfrac{(n-1)y} {2x+\cfrac{(n+1)y} {3nx^{n-1}+\cfrac{(2n-1)y} {2x+\cfrac{(2n+1)y} {5nx^{n-1}+\cfrac{(3n-1)y} {2x+\ddots}}}}}};$

$\sqrt[n]{z}=x+\cfrac{2x\cdot y}{n(2z - y)-y-\cfrac{(1^2n^2-1)y^2}{3n(2z - y)-\cfrac{(2^2n^2-1)y^2}{5n(2z - y)-\cfrac{(3^2n^2-1)y^2}{7n(2z - y)-\ddots}}}}.$

In the case of the fifth root of 34 above (after dividing out selected common factors):

$\sqrt[5]{34} = 2+\cfrac{1} {40+\cfrac{4} {4+\cfrac{6} {120+\cfrac{9} {4+\cfrac{11} {200+\cfrac{14} {4+\ddots}}}}}} =2+\cfrac{4\cdot 1}{165-1-\cfrac{4\cdot 6}{495-\cfrac{9\cdot 11}{825-\cfrac{14\cdot 16}{1155-\ddots}}}}.$

## Complex roots

Every complex number other than 0 has n different nth roots.

### Square roots

The square roots of i

The two square roots of a complex number are always negatives of each other. For example, the square roots of −4 are 2i and −2i, and the square roots of i are

$\frac{1 + i}{\sqrt{2}}\quad\text{and}\quad\frac{-1 - i}{\sqrt{2}}.$

If we express a complex number in polar form, then the square root can be obtained by taking the square root of the radius and halving the angle:

$\sqrt{re^{i\theta}} \,=\, \pm\sqrt{r}\,e^{i\theta/2}.$

A principal root of a complex number may be chosen in various ways, for example

$\sqrt{re^{i\theta}} \,=\, \sqrt{r}\,e^{i\theta/2}$

which introduces a branch cut in the complex plane along the positive real axis with the condition 0 ≤ θ < 2π, or along the negative real axis with −π < θ ≤ π.

Using the first(last) branch cut the principal square root $\sqrt z$ maps $z$ to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software like Matlab or Scilab.

### Roots of unity

The three 3rd roots of 1

The number 1 has n different nth roots in the complex plane, namely

$1,\;\omega,\;\omega^2,\;\ldots,\;\omega^{n-1},$

where

$\omega \,=\, e^{2\pi i/n} \,=\, \cos\left(\frac{2\pi}{n}\right) + i\sin\left(\frac{2\pi}{n}\right)$

These roots are evenly spaced around the unit circle in the complex plane, at angles which are multiples of $2\pi/n$. For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, $i$, −1, and $-i$.

### nth roots

Every complex number has n different nth roots in the complex plane. These are

$\eta,\;\eta\omega,\;\eta\omega^2,\;\ldots,\;\eta\omega^{n-1},$

where η is a single nth root, and 1, ωω2, ... ωn−1 are the nth roots of unity. For example, the four different fourth roots of 2 are

$\sqrt[4]{2},\quad i\sqrt[4]{2},\quad -\sqrt[4]{2},\quad\text{and}\quad -i\sqrt[4]{2}.$

In polar form, a single nth root may be found by the formula

$\sqrt[n]{re^{i\theta}} \,=\, \sqrt[n]{r}\,e^{i\theta/n}.$

Here r is the magnitude (the modulus, also called the absolute value) of the number whose root is to be taken; if the number can be written as a+bi then $r=\sqrt{a^2+b^2}$. Also, $\theta$ is the angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties that $\cos \theta = a/r,$ $\sin \theta = b/r,$ and $\tan \theta = b/a.$

Thus finding nth roots in the complex plane can be segmented into two steps. First, the magnitude of all the nth roots is the nth root of the magnitude of the original number. Second, the direction of a ray from the origin to one of the nth roots involves an angle $\theta / n$ relative to the positive horizontal axis that is[clarification needed] 1/n times the angle $\theta$ of a ray from the origin to the original number relative to the positive horizontal axis. Furthermore, all n of the nth roots are at equally spaced angles from each other.

As with square roots, the formula above cannot be applied consistently to the entire complex plane, but instead leads to a branch cut at the points where θ / n suddenly “jumps”.[clarification needed]

## Solving polynomials

It was once believed that all roots of polynomials could be expressed in terms of radicals and elementary operations; however, while this is true for third degree polynomials (cubics) and fourth degree polynomials (quartics), the Abel-Ruffini theorem (1824) shows that this is not true in general when the degree is 5 or greater. For example, the solutions of the equation

$x^5=x+1\,$

cannot be expressed in terms of radicals. (cf. quintic equation)

For solving any equation of the nth degree numerically, to obtain a result that is arbitrarily close to being exact, see Root-finding algorithm.