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In Mathematics , an ''n''th root of a number ''a'' is a number ''b'', such that ''bn''=''a''. When referring to ''the'' ''n''th root of a Real Number ''a'' it is assumed you are talking about the '''principal ''n''th root''' of the number, which is denotated $\backslash sqrt${Link without Title} {a} with the '''radical''' symbol ($\backslash sqrt\{\backslash ,\backslash ,\}$). The principal square root of a Real Number ''a'' is the unique real number ''b'' that is an n-th root of ''a'' and is of the same sign as ''a''. Note that if ''n'' is even, negative numbers will not have a principal ''n''th root. See Square Root for the case where ''n'' = 2. FUNDAMENTAL OPERATIONS Operations with radicals are given by the following formulas: :$$ \sqrt = \sqrt[n {a} \sqrt[n]{b}, : :$$ \sqrt = \left(\sqrt[n {a} ight)^m = \left(a^{rac{1}{n}} ight)^m = a^{rac{m}{n}}, where ''a'' and ''b'' are positive. For every non-zero Complex Number ''a'', there are ''n'' different complex numbers ''b'' such that ''b''''n'' = ''a'', so the symbol $\backslash sqrt${Link without Title} {a} cannot be used unambiguously. The ''n''th 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 :$a^m\; a^n\; =\; a^\{m+n\}\; \backslash ,$ : :$(a^m)^n\; =\; a^\{mn\}\; \backslash ,$ For example: :$\backslash sqrt${Link without Title} {a^5}\sqrt {Link without Title} {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. :$\backslash sqrt=\; \backslash sqrt[3\{aaaaa\}\; =\; \backslash sqrt=\; a\backslash sqrt[3\{a^2\}$ If you understand how to simplify one radical expression, then addition and subtraction is simply a question about grouping "like terms". For example, :$\backslash sqrt${Link without Title} {a^5}+\sqrt {Link without Title} {a^8} :$=\backslash sqrt${Link without Title} {a^3a^2}+\sqrt {Link without Title} {a^6 a^2} :$=a\backslash sqrt${Link without Title} {a^2}+a^2\sqrt {Link without Title} {a^2} :$=(\{a+a^2\})\backslash sqrt${Link without Title} {a^2} WORKING WITH SURDS Often it is easier to leave the ''n''th roots of numbers unresolved. These unresolved expressions, called surds, can then be manipulated into simpler forms or arranged them to divide each other out. Notationally, the radical symbol ($\backslash sqrt\{\backslash ,\backslash ,\}$) depicts surds, with the upper line above the expression called the Vinculum . A cube root takes the form: :$\backslash sqrt${Link without Title} {a}, which corresponds to , when expressed using indices. All roots can remain in surd form. Basic techniques for working with surds arise from identities. Some basic examples include: $\backslash sqrt\{a^2\; b\}\; =\; a\; \backslash sqrt\{b\}$ $\backslash sqrt\{a\}\; \backslash sqrt\{b\}\; =\; \backslash sqrt\{ab\}$ 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 :$(\backslash sqrt\{a\}+\backslash sqrt\{b\})(\backslash sqrt\{a\}-\; \backslash sqrt\{b\})\; =\; a\; -\; b$, 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 rac{\displaystyle\prod_{k=0}^n (s+t-kt)}{(s+t)n!t^n}x^n

for $k=0,1,2,\backslash ldots,n-1$, where $\backslash sqrt${Link without Title} {a} represents the principle ''n''th root of ''a''.


SOLVING POLYNOMIALS

It was once conjectured that all roots of Polynomial s 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
: $\backslash \; x^5=x+1$
cannot be expressed in terms of radicals.

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


SEE ALSO

• Shifting Nth-root Algorithm

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