nth root

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

rⁿ = x,

where n is the degree of the root. A root of degree 2 is usually called a square root and a root of degree 3, a cube root.

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 number may have many roots of degree n and the roots are not necessarily real numbers.

Roots are usually written using the radical symbol $\sqrt{\,\,}$, with $\sqrt{x}\!\,$ 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, 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; where 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 can be used to prove the Abel-Ruffini theorem; 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, which is the first letter in the word Jadhir "root" in Arabic. (The dh in Jadhir is a voiced interdental fricative, like the th in English the.) Many scholars, however, including Leonhard Euler ^[3] believe it originates from the letter r, the first letter of radix "root" in Latin word 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 asamm (deaf, dumb) for irrational number being translated as surdus (deaf or mute) into Latin. Gherardo of Cremona (c. 1150), Fibonacci (1202) and then Robert Recorde (1551) all used the term to refer to unresolved irrational roots.^[4]

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 x^1/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, 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	√	%E2%88%9A	√
Cube root	∛	U+221B	∛	%E2%88%9B
Fourth root	∜	U+221C	∜	%E2%88%9C

Square roots

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

Main article: Square root

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 = $\sqrt[3]{x}$.

Main article: Cube root

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:

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

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

When simplifying an expression using the exponent form as in x^1/3 normally makes it easier to cancel out powers and roots.

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

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:

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

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

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

If negative roots are allowed or when taking the nth roots of complex numbers problems can occur. 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^[5]

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:

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

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 or rational number. For instance, the fifth root of 34 is

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

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

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

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

In the case of the fifth root of 34 above (after dividing out 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, and along the negative real axis with −π < θ ≤ π.

Roots of unity

The three 3rd roots of 1

Main article: Root of unity

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 2π / 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, ω, ω², ... 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

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

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”.

Solving polynomials

It was once believed that all roots of polynomials could be expressed in terms of radicals and elementary operations; however, the Abel-Ruffini theorem (1824) shows that this is not true in general. 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, see Root-finding algorithm.

See also

• Nth root algorithm
• Shifting nth-root algorithm
• Irrational number
• Algebraic number
• Nested radical
• Twelfth root of two
• Super-root

References

1. ^ Bansal, R K (2006). New Approach to CBSE Mathematics IX. Laxmi Publications. p. 25. ISBN 9788131800133. http://books.google.com/books?id=1C4iQNUWLBwC&pg=PA25#v=onepage&q&f=false. 
2. ^ Silver, Howard A. (1986). Algebra and trigonometry. Englewood Cliffs, N.J.: Prentice-Hall. ISBN 0130212709. 
3. ^ Leonhard Euler (1755) (in Latin). Institutiones calculi differentialis. 
4. ^ "Earliest Known Uses of Some of the Words of Mathematics". Mathematics Pages by Jeff Miller. http://jeff560.tripod.com/s.html. Retrieved 2008-11-30. 
5. ^ McKeague, Charles P. (2011). Elementary algebra. p. 470. http://books.google.com/books?id=etTbP0rItQ4C&printsec=frontcover&dq=editions:q0hGn6PkOxsC&hl=sv&ei=52CsTqv9Go7sOZ_tldAP&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDEQ6AEwAQ#v=onepage&q&f=false. 

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