Nth root

In mathematics, an nth root of a number a is a number b, such that bⁿ=a. When referring to the nth root of a real number a it is assumed you are talking about the principal nth root of the number, which is denotated $\sqrt$ ^*{a} with the radical symbol ( $\sqrt{\,\,}$ ). The principal nth root of a real number a is the unique real number b that is an nth root of a and is of the same sign as a. Note that if n is even, negative numbers will not have a principal nth root. See square root for the case where n = 2.

Fundamental operations

Operations with radicals are given by the following formulas:

\sqrt= \sqrt ^*{b} \qquad a \ge 0, b \ge 0

\sqrt = \frac{\sqrt

^*{b}} \qquad a \ge 0, b > 0

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

where a and b are positive.

For every non-zero complex number a, there are n different complex numbers b such that bⁿ = a, so the symbol $\sqrt$ ^*{a} cannot be used unambiguously. The nth roots of unity are of particular importance.

Once a number has been changed from radical form to exponentiated form, the rules of exponents still apply (even to fractional exponents), namely

a^m a^n = a^{m+n} \,

({\frac{a}{b}})^m = \frac{a^m}{b^m}

(a^m)^n = a^{mn} \,

For example:

\sqrt

^*{a^5}\sqrt^*{a^4} = a^{5/3} a^{4/5} = a^{5/3 + 4/5} = a^{37/15}

If you are going to do addition or subtraction, then you should notice that the following concept is important.

\sqrt = \sqrt

^*{a^3a^2}" target="_blank" >= a\sqrt[3{a^2}

If you understand how to simplify one radical expression, then addition and subtraction is simply a question about grouping "like terms".

For example,

\sqrt

^*{a^5}+\sqrt^*{a^8}

=\sqrt

^*{a^3a^2}+\sqrt^*{a^6 a^2}

=a\sqrt

^*{a^2}+a^2\sqrt^*{a^2}

=({a+a^2})\sqrt

^*{a^2}

Working with surds

Often it is easier to leave the nth roots of numbers unresolved (with radicals visible). These unresolved expressions, called surds, can then be manipulated into simpler forms or arranged them to divide each other out. Notationally, the radical symbol ( $\sqrt{\,\,}$ ) depicts surds, with the upper line above the expression called the vinculum. A cube root takes the form:

\sqrt

^*{a}, which corresponds to

a^{\frac{1}{3}}

, when expressed using indices.

All roots can remain in surd form.

Basic techniques for working with surds arise from identities. Some basic examples include:

$\sqrt{a^2 b} = a \sqrt{b}$

$\sqrt b} = a^{\frac{m}{n}}\sqrt[n {b}$

$\sqrt{a} \sqrt{b} = \sqrt{ab}$

$(\sqrt{a}+\sqrt{b})^{-1} = \frac{1}{(\sqrt{a}+\sqrt{b})} = \frac{\sqrt{a}-\sqrt{b}}{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})} = \frac{\sqrt{a}- \sqrt{b}} {a - b}$

The last of these can serve to rationalize the denominator of an expression, moving surds from the denominator to the numerator. It follows from the identity

(\sqrt{a}+\sqrt{b})(\sqrt{a}- \sqrt{b}) = a - b

which exemplifies a case of the difference of two squares. Variants for cube and other roots exist, as do more general formulae based on finite geometric series.

Infinite series

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

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

with $\ |x|<1$ .

Finding all roots

All the roots of any number, real or complex, may be found with a simple algorithm. The number should first be written in the form ae^iφ (see Euler's formula). Then all the nth roots are given by:

e^{(\frac{\varphi+2k\pi}{n})i} \times \sqrt

^*{a}

for

k=0,1,2,\ldots,n-1

, where

\sqrt

^*{a} represents the principle nth root of a.

Positive real numbers

All the complex solutions of xⁿ = a, or the nth roots of a, where a is a positive real number, are given by the simplified equation:

e^{2\pi i \frac{k}{n}} \times \sqrt

^*{a}

for

k=0,1,2,\ldots,n-1

, where

\sqrt

^*{a} represents the principle nth root of a.

Solving polynomials

It was once conjectured that all roots of polynomials could be expressed in terms of radicals and elementary operations. That this is not true in general is the assertion of the Abel-Ruffini theorem. For example, the solutions of the equation

\ x^5=x+1

cannot be expressed in terms of radicals.

For solving any equation of the n^th degree, see Root-finding algorithm.